CU-BCA-SEM-III-PROBABILITY AND STATICS- Second Draft-converted

Solution: Let A: the card drawn is a queen B: the card drawn is a king C: the card is drawn is a knave(jack) 9.6 SUMMARY • The multiplication rule of probability explains the condition between two events. • For two events A and B associated with a sample space S, the set A∩B denotes the events in which both event A and event B have occurred. The probability of event AB is obtained by using the properties of conditional probability. • Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome. • Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event. 9.7 KEYWORDS • Random Experiment: An experiment, trial, or observation that can be repeated numerous times under the same conditions. • Probability: Some event will happen after many repeated trials 201 CU IDOL SELF LEARNING MATERIAL (SLM)

• Mutually exclusive: A situation where the occurrence of one outcome supersedes the other. • Conditional probability: The probability of one event occurring with some relationship to one or more other events. • Dependent events: The outcome of the first event affects the outcome of the second event. • Independent events: Those events whose occurrence is not dependent on any other event. 9.8 LEARNING ACTIVITY 1. If the events A and B are independent, then prove that the events A and B are also independent. ______________________________________________________________________________ _________________________________________________________________ 2. If the events A and B are independent, then prove that the events A and B are also independent. ______________________________________________________________________________ _________________________________________________________________ 9.9 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. Define the axioms of probability. 2. Prove that probability of the impossible event is zero. 3. If A is the complementary event of A then prove that P ( A ) = 1 - P(A)  1 4. If P(A) = 0.4, P(B) = 0.7 and P(A  B) = 0.3 Find P ( A  B) 5. If the events A and B are independent, then prove that A and B are also independent. 6. State the theorem of total probability 202 CU IDOL SELF LEARNING MATERIAL (SLM)

7. State Bayes’ Theorem. Long Questions 1. A lot consists of 10 good articles, 4 with minor defects and 2 with major defects. Two articles are chosen from the lot at random (without replacement). Find the probability that (i) both are good (ii) both have major defects, (iii) at least 1 is good, (iv) at most 1 is good, (v) exactly 1 is good, (vi) neither has major defects and (vii) neither is good. 2. In a shooting test the probability of hitting the target is ½ for A, 2/3 for B and ¾ for C. If all of them fire at target, find the probability that (i) none of them hits the target and (ii) at least one of them hits the target. 3. An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two balls are drawn at random from the first urn and placed in the second urn and then 1 ball is taken at random from the latter. What is the probability that it is a white ball? 4. A bag contains 5 balls and it is not known how many of them are white. Two balls are drawn from the bag and they are noted to be white. What is the chance that all the balls in the bag are white? 5. A box contains 4 bad and good tubes. Two are drawn out from the box at a time. One of them is tested and found to be good. What is the probability that the other one is also good? 6. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5? 7. A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue? 8. From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings? B. Multiple choice Questions 1. In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green? b. 1/3 c. ¾ d. 7/19 e. 9/21 203 CU IDOL SELF LEARNING MATERIAL (SLM)

2. What is the probability of getting a sum 9 from two throws of a dice? a. 1/6 b. 1/8 c. 1/9 d. 1/12 3. Three unbiased coins are tossed. What is the probability of getting at most two heads? a. 3/4 b. 1/4 c. 3/8 d. 7/8 4. A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is: a. 1/22 b. 3/22 c. 2/91 d. 2/77 5. If A and B are mutually exclusive then a. P(A∪B) =P(A)+P(B) b. P(A∪B) =P(A)-P(B) c. P(A∪B) =P(A). P(B) d. P(A)=P(B) Answers 1.a, 2.c, 3.d, 4.c, 5.a 204 CU IDOL SELF LEARNING MATERIAL (SLM)

9.10 REFERENCES Reference Books: • Dr. B. Krishna Gandhi, Dr. T.K.V Iyengar, M.V.S.S.N. Prasad, Probability and Statistics, S. Chand Publishing Co. • Quantitative Methods for Business & Economics by Mouhammed, Publisher: PHI, 2007 Edition. • Statistical Methods by S.P Gupta, Publisher: Sultan Chand & Sons, 2008 Edition. • Research Methodology by C. R. Kothari, Publisher: Vikas Publishing House Textbooks: • S.C. Gupta, V.K. Kapoor, Fundamental of Mathematical Statistics, Sultan Chand and Company. • Research Methodology and Statistical Methods by T. Subbi Reddy, Publisher: Reliance Publishing House • Introduction to Linear Optimization,\" by Dimitris Bertsimas and John 205 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 10: PROBABILITY, RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION Structure 10.0 Learning Objectives 10.1 Introduction 10.2 Independent and Dependent Events 10.3 Total probability 10.4 Bayes' Theorem 10.5 Random variable and its probability distribution 10.6 Summary 10.7 Keywords 10.8 Learning Activitiy 10.9 Unit End Questions 10.10 References 10.0 LEARNING OBJECTIVES After studying this unit students will be able to • Explain the notion of discrete random variables. • Learn how to use discrete random variables to solve probabilities of outcomes. • State the characteristics about continuous random variables and probability distributions • Explain the relation between a normal random variable and a standard normal random variable • Calculate the expectation of a function of a discrete random variable. • Calculate the variance of a discrete random variable. 10.1 INTRODUCTION Two or more events are said to be independent, when the outcome of one does not affect the other. The results obtained by throwing a dice are independent of the results obtained by drawing 206 CU IDOL SELF LEARNING MATERIAL (SLM)

an ace from a pack of cards. To consider two events that is not independent let A stand for a firm's spending a large amount of money on advertisement and B for it showing an increase in sales. Of course, advertising does not guarantee higher sales, but the probability that the firm will show an increase in sales will be higher if A has taken place. Dependent events are those in which the occurrence or non-occurrence of one event in any one trial affects the probability of other events in other trials. For instance, if we draw a card from a pack of playing cards and is not replaced then this will alter the probability that the second card drawn is an ace, Similarly, the probability of drawing a queen from a pack of 52 cards is 4 / 52 or l /13; But if the card drawn (queen) is not replaced in the pack, the probability of drawing again a queen is; 1 (the pack now contains only 51 cards out of which there are 3 queens). Bayes' Theorem is a theorem of probability originally stated by the Reverend Thomas Bayes. Its existence can be seen in a way of understanding how the probability is affected by a new piece of evidence. It has been used in wide variety of contexts, ranging from marine biology to the development of \"Bayesian\" spam blockers for email systems. In the philosophy of science, it has been used to try to clarify the relationship between theory and evidence. Many insights in the philosophy of science involving confirmation, falsification, the relation between science and pseudoscience, and other topics that can be made more precise, sometimes they can be extended or corrected with the use of Bayes' theorem. 10.2 INDEPENDENT AND DEPENDENT EVENTS In Probability, the set of outcomes of an experiment is called events. There are different types of events such as independent events, dependent events, mutually exclusive events, and so on. If the probability of occurrence of an event A is not affected by the occurrence of another event B, then A and B are said to be independent events. Example: Consider a rolling a die. If A is the event ‘the number appearing is odd’ and B be the event ‘the number appearing is a multiple of 3’, then P(A)= 3/6 = 1/2 and P(B) = 2/6 = 1/3 Also, A and B is the event ‘the number appearing is odd and a multiple of 3’ so that P (A ∩ B) = 1/6 P(A│B) = P (A ∩ B)/ P(B) = (1/6)/ (1/3) = 1/2 10.3 TOTAL PROBABILITY Law of Total Probability 207 CU IDOL SELF LEARNING MATERIAL (SLM)

For two events A and B associated with a sample space S, the sample space can be divided into a set A ∩ B′, A ∩ B, A′ ∩ B, A′ ∩ B′. This set is said to be mutually disjoint or pairwise disjoint because any pair of sets in it is disjoint. Elements of this set are better known as a partition of sample space. This can be represented by the Venn diagram as shown in fig. 1. In cases where the probability of occurrence of one event depends on the occurrence of other events, we use total probability theorem. Statement: Let events C1, C2 . . . Cn form partitions of the sample space S, where all the events have a non-zero probability of occurrence. For any event, A associated with S, according to the total probability theorem, Example 1: A person has undertaken a mining job. The probabilities of completion of job on time with and without rain are 0.42 and 0.90 respectively. If the probability that it will rain is 0.45, then determine the probability that the mining job will be completed on time. Solution: 208 CU IDOL SELF LEARNING MATERIAL (SLM)

Let A be the event that the mining job will be completed on time and B be the event that it rains. We have, P(B) = 0.45, P (no rain) = P(B′) = 1 − P(B) = 1 − 0.45 = 0.55 By multiplication law of probability, P(A|B) = 0.42 P(A|B′) = 0.90 Since, events B and B′ form partitions of the sample space S, by total probability theorem, we have P(A) = P(B) P(A|B) + P(B′) P(A|B′) =0.45 × 0.42 + 0.55 × 0.9 = 0.189 + 0.495 = 0.684 So, the probability that the job will be completed on time is 0.684. 10.4 BAYES' THEOREM Statement Let E1, E2…, En be a set of events associated with a sample space S, where all the events E1, E2…, En have nonzero probability of occurrence and they form a partition of S. Let A be any event associated with S, then according to Bayes theorem, for any k = 1, 2, 3, …., n Example 1: A bag I contain 4 white and 6 black balls while another Bag II contains 4 white and 3 black balls. One ball is drawn at random from one of the bags, and it is found to be black. Find the probability that it was drawn from Bag I. Solution: Let E1 be the event of choosing the bag I, E2 the event of choosing the bag II, and A be the event of drawing a black ball. Then, P(E1) = P(E2) = 1/2 Also, P(A|E1) = P (drawing a black ball from Bag I) = 6/10 = 3/5 P(A|E2) = P (drawing a black ball from Bag II) = 3/7 By using Bayes’ theorem, the probability of drawing a black ball from bag I out of two bags, 209 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 2: A man is known to speak truth 2 out of 3 times. He throws a die and reports that the number obtained is a four. Find the probability that the number obtained is actually a four. Solution: Let A be the event that the man reports that number four is obtained. Let E1 be the event that four is obtained and E2 be its complementary event. Then, P(E1) = Probability that four occurs = 1/6 P(E2) = Probability that four does not occurs = 1 – P(E1) = 1 −1/6 = 5/6 Also, P(A|E1) = Probability that man reports four and it is actually a four = 2/3 P(A|E2) = Probability that man reports four and it is not a four = 1/3 By using Bayes’ theorem, probability that number obtained is actually a four, 10.5 RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION Random Variable: A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment's outcomes. Random variables are often designated by letters and can be classified as discrete, which are variables that have specific values, or continuous, which are variables that can have any values within a continuous range. Example 1: Suppose you simultaneously toss two fair coins. Let X be the number of heads observed. Find the probability associated with each value of the random variable X. Since there are two coins, and each coin can be either heads or tails, there are four possible outcomes (HH, HT, TH, TT), each with a probability of 14. Since X is the number of heads observed, x=0,1,2. 210 CU IDOL SELF LEARNING MATERIAL (SLM)

We can identify the probabilities of the simple events associated with each value of X as follows: P(x=0) P(x=1) P(x=2) =P(TT)=14=P(HT)+P(TH)=14+14=12=P(HH)=14 This is a complete description of all the possible values of the random variable, along with their associated probabilities. We refer to this as a probability distribution. This probability distribution can be represented in different ways. Sometimes it is represented in tabular form and sometimes in graphical form. Both forms are shown below. In tabular form: x P(x) 0 14 1 12 2 14 Table 10.1: The tabular form of the probability distribution for the random Example 2: Solution: 211 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 3: 212 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example:4 Solution: 214 CU IDOL SELF LEARNING MATERIAL (SLM)

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Probability distribution A probability distribution of a random variable specifies the values the random variable can assume, along with the probability of it assuming each of these values. All probability distributions must satisfy the following two conditions: a. P(x) ≥ 0, for all values of X b. ∑P(x) = 1, for all values of X A probability distribution for a discrete random variable is a mutually exclusive listing of all possible numerical outcomes for that random variable such that a particular probability of 216 CU IDOL SELF LEARNING MATERIAL (SLM)

occurrence is associated with each outcome. In case of a fair six-faced dice that will not stand on edge or roll out of sight (null events) the probability distribution for the outcomes of single roll of the dice shall be as follows: Figure 10.1: Probability Distribution Since all possible outcomes are included, this listing is complete (or collectively exhaustive) and thus be probabilities must sum up to 1. From the above table we can obtain various probabilities for the rolling of a ' fair dice. For example, Figure 10.2 Example of Probability Distribution It should be noted that a random variable is a numerical quantity whose value is determined by the outcome of a random (chance) experiment. When a random experiment is performed the totality of outcomes of the experiment forms a set which is called 'sample space' [S] of the 217 CU IDOL SELF LEARNING MATERIAL (SLM)

experiment. Let the random experiment be tossing of a coin 2 times. Here S== [(T, T), (T, H), (H, T), (H, H)]. If we replace T by 0 and H by 1 then the number of heads obtained in both the trials shall be: (T, T) - 0 (T, H) -1 (H, T) - 1 (H, H) ~ 2 The sample space S can be written as: [0, 1 2,] and here p (X = 0) = P (T, T) = ¼ P (X = 1) = P [(T, H), (H, T)] = 1/2 P (X = 2) = P [(H, H)] = 1/4 Hence ∑P(X) = 1/4 + 1/2 + 1/4 = 1. Such a function P (X) is called the 'probability function' of the random variable X The probability distribution is the outcome of the different probabilities taken by selection the random variable X. A random variable can be either distinct or unbroken. A random variable is supposed to be discrete if the set of values defined by it over the sample space is finite. On the other hand, a random variable is 'continuous' if it can assume any (real) value in an interval. If the random variable X is a discrete one, the probability function P (X) is called 'probability mass function' and its distribution as 'discrete probability distribution' and if the random variable X is of continuous type, then the probability function f(X) is called probability density function and its distribution is called continues probability distribution. Knowledge of the expected behaviour of a phenomenon or, in other words, the expected frequency distribution is of great help in a large number of problems in practical life. They serve as benchmarks against which to compare observed distributions and act as substitutes for actual distributions when the latter are costly to obtain or cannot be obtained at all. They provide decision-makers with a logical basis for making decisions and are useful in making predictions on the basis of limited information or theoretical considerations. For example, the proprietor of a shoe store must know something about the distribution of the size of his potential customers' feet: otherwise, he may find himself with huge stock of shoes which have no market. Similarly, the manufacturer of ready-made garments must know the sizes of collars for which he expects maximum demand so that he has no stock of unwanted sizes. In a similar way the teachers in the school, college or university should know what they expect of the students. It is only then that 218 CU IDOL SELF LEARNING MATERIAL (SLM)

they would be in a position to comment on good or bad performance. Amongst theoretical or expected frequency distributions, the following are more popular: • Binomial Distribution, • Poisson distribution, • Normal Distribution. 10.6 SUMMARY • In statistics, independent events are two events wherein the occurrence of one event does not affect the occurrence of another event or events. • Total probability can be calculated using the known probabilities of several distinct events. • A random variable is a numerical description of the outcome of a statistical experiment. • The probability distribution for a random variable describes how the Probabilities are distributed over the values of the random variable. 10.7 KEYWORDS • Random Variable(X): A set of possible finite or infinite values from a random experiment. • Mean: The expected value or expectation of X and denoted by E(X). • Variance V(X): The product of the square of the difference between the value of the random variable • P(X): The Probability of random variables 10.8 LEARNING ACTIVITY 1.A doctor is called to see a sick child. The doctor knows (prior to the visit) that 90% of the sick children in that neighbourhood are sick with the flu, denoted by F, while 10% are sick with the measles, denoted by M. A well-known symptom of measles is a rash, denoted by R. The probability of having a rash for a child sick with the measles is 0.95. However, occasionally children with the flu also develop a rash, with conditional probability 0.08. Upon examination the child, the doctor finds a rash. Then what is the probability that the child has the measles? ______________________________________________________________________________ _________________________________________________________________ 219 CU IDOL SELF LEARNING MATERIAL (SLM)

10.9 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. A bag contains 5 red and 5 black balls. A ball is drawn at random, its color is noted, and again the ball is returned to the bag. Also, 2 additional balls of the color drawn are put in the bag. After that, the ball is drawn at random from the bag. What is the probability that the second ball drawn from the bag is red? 2. State the theorem of total probability. 3. Define Random Variable. 4. A random variable X has the following probability function X 01234 P(X) K 3K 5K 7K 9K Find the value K 5. If the random variable X takes the values 1,2,3,4 such that 2P(X=1) =3P(X=2) =P(X=3) =5P(X=4). Find the probability distribution and cumulative distribution function of X. Long Questions 1. Of the students in the college, 60% of the students reside in the hostel and 40% of the students are day scholars. Previous year result reports that 30% of all students who stay in the hostel scored A Grade and 20% of day scholars scored A grade. At the end of the year, one student is chosen at random and found that he/she has an A grade. What is the probability that the student is a hostlier? 2. From the pack of 52 cards, one card is lost. From the remaining cards of a pack, two cards are drawn and both are found to be the diamond cards. What is the probability that the lost card being a diamond? 3. A bag contains 5 balls and it is not known how many of them are white. Two balls are drawn from the bag and they are noted to be white. What is the chance that all the balls in the bag are white? 4. A bolt is manufactured by 3 machines A, B, C. A turns out twice as many items as B, and machine B and C produce equal number of items.2% of bolts produced by A and B are 220 CU IDOL SELF LEARNING MATERIAL (SLM)

defective and 4% of bolts produced by C are defective. All bolts are put in one stock pile and one is chosen from this pile. What is the probability that it is defective? 5. A random variable X has the following probability function X 0 1 2 34 5 6 7 8 P(x) a 3a 5a 7a 9a 11a 13a 15a 17a (i) Determine value of ‘a’ (ii) Find P(x<3), P(x  3), P(0<x<5) (iii)Find the distribution function of x. 6. The probability mass function of a discrete R. V X is given in the following table: X -2 -1 0 1 2 3 P(X=x) 0.1 k 0.2 2k 0.3 3k Find (i) the value of k, (ii) P(X<1), (iii) P (-1< X ≤ 2) B. Multiple choice Questions 1. Two fair coins are flipped. As a result of this, tails and heads run occurred where a tail run is a consecutive occurrence of at least one head. Determine the probability function of number of tail runs. a. ½ b. 5/6 c. 32/19 d. 6/73 2. Two t-shirts are drawn at random in succession without replacement from a drawer containing 5 red t-shirts and 8 white t-shirts. Find the probabilities of all the possible outcomes. a. 1 b. 13 c. 40 d. 346 3. A meeting has 12 employees. Given that 8 of the employees is a woman, find the probability that all the employees are women? 221 CU IDOL SELF LEARNING MATERIAL (SLM)

a. 11/23 b. 12/35 c. 2/9 d. 1/8 4. If P(A∩B) = 0.2 and P(A) = 0.8 then what is the value of P(B/A) a. 0.2 b. 0.25 c. 0.3 d. 0.35 5. What is the sample size for three coins? a. 2 b. 4 c. 16 d. 8 Answers 1.a, 2.a, 3.c, 4.b, 5.d 10.10 REFERENCES Reference Books: • Quantitative Methods for Business & Economics by Mouhammed, Publisher: PHI, 2007 Edition. • Quantitative Techniques for Managerial Decisions by A. Sharma, Publisher: Macmillan, 2008 Edition. • Quantitative Techniques for Decision Making by A. Sharma, Publisher: HPH, 2007 Edition. Textbooks: • S.C. Gupta, V.K. Kapoor, Fundamental of Mathematical Statistics, Sultan Chand and Company. • Seymour Lipschutz, Jack Schiller, Jack Schiller S, Introduction to Probability & Statistics, McGraw-Hill Publishers. 222 CU IDOL SELF LEARNING MATERIAL (SLM)

• Research Methodology and Statistical Techniques by Santosh Gupta, Publisher: Deep and Deep Publication • Research Methodology and Statistical Methods by T. Subbi Reddy, Publisher: Reliance Publishing House 223 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 11: MEAN AND VARIANCE OF HAPHAZARD VARIABLE Structure 11.0 Learning Objectives 11.1 Introduction 11.2 Mean and variance of haphazard variable Distribution 11.3 Bernoulli Distribution 11.4 Binomial Distribution 11.5 Summary 11.6 Keywords 11.7 Learning activity 11.8 Unit End Questions 11.9 References 11.0 LEARNING OBJECTIVES After studying this unit students will be able to: • Explain the definition of the mean, or expected value, of a discrete random variable. • State the definition of the standard deviation of a discrete random variable. • Outline the definition of the variance of a discrete random variable. • Describe the expected value of a variable. • Explain the normal curve and the concept behind two-thirds of cases being within one standard deviation of the mean. • Calculate the upper and lower boundaries of the typical range of scores in a normal distribution based on the mean and the standard deviation. 11.1 INTRODUCTION In probability and statistics, we can find out the average of a random variable. The term average is the mean or the expected value or the expectation in probability and statistics. Once we have 224 CU IDOL SELF LEARNING MATERIAL (SLM)

calculated the probability distribution for a random variable, we can calculate its expected value. Mean of a random variable shows the location or the central tendency of the random variable. There are some classic random variable abstractions that show up in many problems. At this point in the class, you will learn about several of the most significant discrete distributions. When solving problems, if you are able to recognize that a random variable fit one of these formats, then you can use its pre calculated probability mass function (PMF), expectation, variance, and other properties. Random variables of this sort are called “parametric” random variables. If you can argue that a random variable fall under one of the studied parametric types, you simply need to provide parameters. A good analogy is a class in programming. Creating a parametric random variable is very similar to calling a constructor with input parameters. 11.2 MEAN AND VARIANCE OF HAPHAZARD VARIABLE Mean The expectation or the mean of a discrete random variable is a weighted average of all possible values of the random variable. The weights are the probabilities associated with the corresponding values. It is calculated as, E(X) = μ= Σi xi pi i = 1, 2, …, n E(X) = x1p1 + x2p2 + … + xnpn. Properties of Mean of Random Variables • If X and Y are random variables, then E (X + Y) = E(X) + E(Y). • If X1, X2, …, Xn are random variables, then E (X1 + X2 + … + Xn) = E(X1) + E(X2) + … + E(Xn) = Σi E(Xi). • For random variables, X and Y, E(XY) = E(X) E(Y). Here, X and Y must be independent. • If a is any constant and X is a random variable, E[aX] = a E[X] and E [X + a] = E[X] Variance The variance of a random variable shows the variability or the scatterings of the random variables. It shows the distance of a random variable from its mean. It is calculated as σx2 = Var (X) = ∑i (xi − μ)2 p(xi) = E (X − μ)2 or, Var(X) = E(X2) − [E(X)]2. E(X2) = ∑i xi2 p(xi), and [E(X)]2 = [∑i xi p(xi)]2 = μ2. If the value of the variance is small, then the values of the random variable are close to the mean. Properties of Variance of Random Variables 225 CU IDOL SELF LEARNING MATERIAL (SLM)

• The variance of any constant is zero i.e., V(a) = 0, where a is any constant. • If X is a random variable, and a and b are any constants, then V (aX + b) = a2 V(X). • For any pair-wise independent random variables, X1, X2, …, Xn and for any constants a1, a2, …, an; V (a1X1 +a2 X2 + … +anXn) = a12 V(X1) + a22 V(X2) + … + an2 V(Xn). Example: Calculate the mean and variance for a random variable, X defined as the number of tails in four tosses of a coin. Also, draw the probability distribution. Solution: Let T represents a tail and H, a head. X denotes the number of tails in four tosses of a coin. X takes the value 0, 1, 2, 3, 4. S. No. Possible Number of S. No. Possible Number of outcomes Tails, X outcomes Tails, X 1 THTH 2 9 THTT 3 2 HHTH 1 10 HHTT 2 3 TTTH 3 11 TTTT 4 4 HTTH 2 12 HTTT 3 5 THHH 1 13 THHT 2 6 HHHH 0 14 HHHT 1 7 TTHH 2 15 TTHT 3 8 HTHH 1 16 HTHT 2 226 CU IDOL SELF LEARNING MATERIAL (SLM)

P (X = 0) = 1⁄16, P (X = 1) = 4⁄16 = 1⁄4, P (X = 2) = 6⁄16 = 3⁄8, P (X = 3) = 4⁄16 = 1⁄4, P (X = 4) = 1⁄16 The probability distribution of X is x0123 4 p(x) 1⁄16 1⁄4 3⁄8 1⁄4 1⁄16 E(X) = Σi xipi = 1 × 1⁄4 + 2 × 3⁄8 + 3 × 1⁄4 + 4 × 1⁄16 = 8⁄4 = 2. E(X2) = 12 × ¼ + 22 × 3⁄8 + 32 × ¼ + 42 × 1⁄16 = ¼ + 3⁄2 + 9⁄4 + 1 = 5. So, Variance of X = V(X) = E(X2) – [E(X)]2 = 5 – 22 = 1. 11.3 BERNOULLI DISTRIBUTION In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability p = 1- p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is 227 CU IDOL SELF LEARNING MATERIAL (SLM)

success with probability p and failure with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent \"heads\" and \"tails\" (or vice versa), respectively, and p would be the probability of the coin landing on heads or tails, respectively. The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two- point distribution, for which the possible outcomes need not be 0 and 1. Properties: Mean: The expected value of a Bernoulli random variable X is Variance: The variance of a Bernoulli distributed X is Example 1: 228 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example 2: 229 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 230 CU IDOL SELF LEARNING MATERIAL (SLM)

11.4 BINOMIAL DISTRIBUTION The binomial distribution also known as 'Bernoulli Distribution' is associated with the name of a Swiss mathematician James Bernou11i also known as Jacques or Jakob (1654-1705). Binomial distribution is a probability distribution expressing the probability of one set of dichotomous alternatives which can either be success or failure. This type of distribution is used to describe a wide variety of processes in business and the social sciences. There is certain type of process which gives rise to this distribution is usually referred to as Bernoulli trial or as a Bernoulli process. The mathematical model for a Bernoulli process is developed under a very specific set of assumption involving the concept of a series of experimental trials. These assumptions are: • For fixed number of trials, we perform a number of experiments under the similar type of conditions. • In each trial there are only two possible outcomes of the experiment. For lack of a better nomenclature, they are called \"success\" or \"failure\". The sample space of possible outcomes on each experimental trial is shown by :S = {failure, success} • The probability of a success is denoted by p which remains constant for all trials. The probability of a failure is denoted by q, q is equal to (1 - p). If the probability of success is not the same in all trials, then we will not have binomial distribution. If 5 balls are randomly picked up from an urn containing 20 white and 30 red balls. This is a binomial experiment only when each ball is replaced before another ball is drawn. If we draw another ball without replacement then it is not binomial experiment • Statistically, the trials are independent (the outcomes of any trial or sequence of trials do not affect the outcomes of subsequent trials). This model is useful to answer questions such as this: If we conduct experiment n times under the fixed conditions then what is the probability of getting exactly x successes? If we toss 10 coins together then what is the probability of obtaining exactly two heads? How binomial distribution arises can be seen from the following: If a coin is tossed once there are two outcomes namely tail or head. The probability of obtaining a head or p = 1/2 and the probability of obtaining a tail q = ½, these are terms of the binomial (q + p). Similarly, if two coins are tossed simultaneously there are four possible outcomes: 231 CU IDOL SELF LEARNING MATERIAL (SLM)

Since by expanding the binomial (q + p) n, we obtain probability of 0, 1, 2 ... nheads, the probability distribution is naturally called the Binomial Probability Distribution or simply the Binomial Distribution. The general form of the distribution is: Example 1: 232 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 2: Example 3: 233 Solution: CU IDOL SELF LEARNING MATERIAL (SLM)

Example 4: Solution: 234 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 5: X is a random variable following Binomial distribution with mean 2.4 and variance 1.44. Find P( X  5) and P(1  X  4). Solution: Let X  B(n, p). Then PX = x = ncx p xq n−x , x = 0,1,2,3....n Mean = np and variance = npq np =2.4 and npq =1.44  q = npq = 1.44 = 0.6 np 2.4 p = 1 − q  p = 0.4and n = 2.4 = 6 0.4 Hence PX = x = 6cx (0.4) x (0.6)6−x , x = 0,1,2...6 235 CU IDOL SELF LEARNING MATERIAL (SLM)

PX  5 = PX = 5+ PX = 6 = 6c5 (0.4)5 (0.6) + 6c6 (0.4)6 =0.0369+0.0041 PX  5 = 0.0410 P1  X  4 = PX = 2+ PX = 3+ PX = 4 = 6c2 (0.4)2 (0.6)4 + 6c3 (0.4)3 (0.6)3 + 6c4 (0.4)4 (0.6)2 =0.31104+0.27648+0.13824 P1  X  4 = 0.7258 Example 6: A perfect cube is thrown a large number of times in sets of 8. The occurrence of 2 or 4 is called a success. In what proportion of sets would you expect 3 successes? Solution: P2 or 4 = 2 = 1 63 Hence p = 1 , q = 2 and n = 8 33  1  3  2  5  3   3  PX = 3 = 8C 3 = 0.2730 In nearly 27.3 percent of cases, one would expect Successes. Example 7: A and B play a game in which their chances of winning are in the ratio 3:2 find A’s chance of winning at least 3 games out of 5 games played. Solution: Probability for A’s winning = 3 5 Probability for B’s winning = 2 5 236 CU IDOL SELF LEARNING MATERIAL (SLM)

Let X: number of games in which A wins PX = r = ncr pr qn−r  3  r  2  5−r  5   5  = 5c r P (at least 3 games out of 5 for A’s winning) = PX  3=P (3) +P (4) +P (5)  3 3  2 2  3  4  2   3 5  5  5  5   5   5 = 5C3 + 5C 4 + 5C5 = 1 10  27  4 + 5 81 2 + 243 55 =0.6823 Example 8: Six dice are thrown 729 times. How many times do you expect at least 3 dice show a 5 or 6? Solution: Let X: the number of the times the dice shown 5 or 6 P 5or 6= 1 + 1 = 1 66 3  p = 1 and q = 2 33 Here n=6 To evaluate the frequency of X  3. By Binomial Theorem,  1  r  2  6−r  3   3  PX = r= 6c r Where r=0, 1, 2…6 PX  3 = P(3) + P(4) + P(5) + P(6) = 6c3  1 3  2 3 + 6c4  1 4  2  2 + 6c5  1 5  2  + 6C6  1  6  3  3   3  3   3   3   3  =0.3196 237 CU IDOL SELF LEARNING MATERIAL (SLM)

 Expected number of times at least 3 dice to show 5 or 6 =N PX  3 =729 0.3196 =233 Example 9: The probability of a man hitting a target is 1 (i)if he fires 7 times, what is the probability of his 4 hitting the target at least twice? (ii) How many times must he fire so that the probability of hitting the target at least once is greater than? 2 3 Solution: Probability of hitting the target p=1 q = 1 − p = 3 4 4 Let denote the number of times of hitting the target, here n=7 By binomial distribution,  1  r  3  7−r  4   4  PX = r= 7cr , r = 0,1,2,...7 (i) P [hitting the target at least twice] = PX  2 = 1− PX  2 = 1− P(0) + p(1) = 1−  3  7 + 7 1  3 6    4   4  4   = 1 − 10  36 = 0.5551(app) 47 (ii) P [ hitting the target at least once] = 1− PX = 0 = 1− qn 238 CU IDOL SELF LEARNING MATERIAL (SLM)

1 −q n  2  1 −  3 n  2 3 4 3 1 − 2   3 n 3 4  3 n  1 4 3 n=4 Importance of the Binomial Distribution The binomial probability distribution is a discrete probability distribution that is useful in describing an enormous variety of real-life events. For example, a quality control head wants to know the probability of faulty light bulbs in a random sample of 10 bulbs if 10 per cent of the bulbs are defective. He can quickly obtain the answer from tables of the binomial probability distribution. The binomial distribution can be used in following situations: • The outcome or results of each trial in the process is one out of two types of possible outcomes. We can say that they are attributes. • The possibility of outcome of any trial does not change and is independent of the results of previous trials. The following examples will illustrate the applications of binomial distribution. 11.5 SUMMARY • A mean refers to the average or the most common value in a collection of, or average. • A large variance indicates that the numbers are further spread out. • A small variance indicates a small spread of numbers from the mean. • The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial. • Binomial distribution summarizes the number of trials, or observations when each trial has the same probability of attaining one particular value. 11.6 KEYWORDS • Probability distribution: A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. 239 CU IDOL SELF LEARNING MATERIAL (SLM)

• Random Variable(X): A set of possible finite or infinite values from a random experiment. • Mean: The expected value or expectation of X and denoted by E(X). • Variance V(X): The product of the square of the difference between the value of the random variable • P(X): The Probability of random variables. 11.7 LEARNING ACTIVITY Students will be learning the mean and variance of random variables and will be encouraged to work together to get a class average. Students are introduced to statistics and why it is important to daily life. A random variable X has the following probability function X 0 1 2 34 5 6 7 8 P(x) a 3a 5a 7a 9a 11a 13a 15a 17a (iv)Determine value of ‘a’ (v) Find P(x<3), P(x  3), P(0<x<5) (vi)Find the distribution function of x. (vii) P(2x+3)>5 (viii) P(2≤x≤4.5) / P(x≥1) (ix)Find the minimum value of x if P(X≤x) > 0.5 P (x=3or x=5) ______________________________________________________________________________ _________________________________________________________________ 11.8 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. What is the relationship between probability and statistics? 2. What is the probability formula? 240 CU IDOL SELF LEARNING MATERIAL (SLM)

3. What is the example of statistics? 4. What is the meaning of probability in statistics? 5. The mean and variance of the binomial distribution are 4 and 5 respectively. Find P(x=0). Long Questions 1. A discrete random variable has the following probability function X: 1 2 3 4 5 P(x): 2/10 1/10 3/10 1/10 3/10 Find P (X= 3) 2. If E(X) = E (X 2) = 0.5 then what is the value of Var(X). 3. The probability function is given by P(x=r) = kr3, r =1,2,3 &4. Find (i) the value of k (ii) Distribution function (iii) Mean and variance. 4. The probability function of an infinite discrete distribution is given by P(X = j) = 1 , j = 1,2,3...... Find (i) Mean of X (ii) P (X is even) (iii) P( X  5) and (iv) 2j P (X is divisible by 3). 5. A and B play a game in which their chances of winning are in the ratio 3:2. Find A’s chance of winning at least 3 games out of 5 games played. 6. Six dice are thrown 729 times. How many times do you expect at least 3 dice show a 5 or 6? 7. The probability of a man hitting a target is ¼. (i) If he fires 7 times, what is the probability of his hitting the target at least twice? (ii) How many times must he fire so that the probability of hitting the target at least once is greater than 2/3? B. Multiple choice Questions 1. Let X is denoted as the number of heads in three tosses of a coin. Determine the mean and variance for the random variable X. a. 4.8 b. 6 c. 3.2 d. 1.5 2. In a Binomial Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by ___________ a. Np b. n c. P 241 CU IDOL SELF LEARNING MATERIAL (SLM)

d. np(1-p) 3. In a Binomial Distribution, if p, q and n are probability of success, failure and number of trials respectively then variance is given by ___________ a. Np b. Npq c. np2q d. npq2 4. If ‘X’ is a random variable, taking values ‘x’, probability of success and failure being ‘p’ and ‘q’ respectively and ‘n’ trials being conducted, then what is the probability that ‘X’ takes values ‘x’? Use Binomial Distribution a. P (X = x) = nCx px qx b. P (X = x) = nCx px q(n-x) c. P (X = x) = xCn qx p(n-x) d. P (x = x) = xCn pn qx 5. If X = 0,1,2,3,4, and P(x) = k, 3k,5k,7k,9k then what is the value of 'k' a. 0.4 b. 0.5 c. 0.2 d. 0.3 Answers 1.d, 2.a, 3.b, 4.b, 5.a 11.9 REFERENCES Reference Books: • Quantitative Methods for Business & Economics by Mouhammed, Publisher: PHI, 2007 Edition. • Quantitative Techniques for Managerial Decisions by A. Sharma, Publisher: Macmillan, 2008 Edition. • Quantitative Techniques for Decision Making by A. Sharma, Publisher: HPH, 2007 Edition. 242 CU IDOL SELF LEARNING MATERIAL (SLM)

• Statistical Methods by S.P Gupta, Publisher: Sultan Chand & Sons, 2008 Edition. • Research Methodology by C. R. Kothari, Publisher: Vikas Publishing House Textbooks: • S.C. Gupta, V.K. Kapoor, Fundamental of Mathematical Statistics, Sultan Chand and Company. • Seymour Lipschutz, Jack Schiller, Jack Schiller S, Introduction to Probability & Statistics, McGraw-Hill Publishers. • Research Methodology and Statistical Techniques by Santosh Gupta, Publisher: Deep and Deep Publication • Research Methodology by V. P. Pandey, Publisher: Himalaya Publication • Research Methodology in Management by Arbind and Desai, Publisher: Ashish Publication House 243 CU IDOL SELF LEARNING MATERIAL (SLM)