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5.2 MEAN, VARIANCE AND EXPECTATION Mean xPx The mean of a discrete random variable is also called its expected value and is denoted by E(X). The mean or expected value is the value that we expect to observe on average, if the experiment is repeated many times. For example, we may expect a car salesperson to sell, on average, 2.4 cars per week. This does not mean that every week this salesperson will sell exactly 2.4 cars. If we observe for many weeks, this salesperson will sell a different number of cars during different weeks. However, the average for all these weeks will be 2.4 cars per week. Expectation E X xPx Variance Var X 2 x Px 2 2 Standard deviation x Px 2 2 Example 1 If three coins are tossed. Find the mean, variance and standard deviation of the number of tails that occur. Example 2 The probability distribution of the discrete random variable X as follows X 1 P(X) 0.4 Find E(X). 2 c 3 4 0.1 0.09 5 0.06 Example 3 A discrete random variable X takes value 0, 1 and 2 with probability P0 , P1 and P2 respectively. Given that E X 4 5 and Var X . Find the value of P0 , P1 and P2 . 3 9 Example 4 The probability that Sofia will win a game is 0.01. So the probability she will not win is 0.99. If Sofia wins, she will be given RM100, while if she loses, she must pay RM5. If X represents amount of Sofia win (or loses), what the expected value of X. Example 5 Let X be the number of cars that Azura sells on a typical Saturday. The probability distribution for X is given below Car Sold, X Probability 0 0.15 1 0.4 2 0.25 3 0.15 4 0.05 (i) Find the probability that on a typical Saturday, Azura will sell at least 2 cars? (ii) Find the mean and standard deviation of X. (iii)If Azura is paid RM50 per day and earns a commission of RM250 for each car she sells, what is her expected earnings on a typical Saturday? Exercise 5.2 1. A random variable X has the following probability distribution. Find the expected value and standard deviation of x. x P(X = x) 1 0.12 3 0.23 5 0.16 7 9 0.35 0.14 2. The following table gives the probability distribution of the number of breakdowns per month of a computer. Breakdowns Per month 0 1 2 3 Probability .20 0.35 0.30 0.15 (a) Find the expected number of breakdowns per month of the computer. Interpret your result. (b) Find the variance and the standard deviation. 3. Given the following probability distribution of x. Compute the mean, variance and standard deviation of x (a) x 2 4 P(x) 0.2 0.15 x -2 6 8 0.45 0.2 -0.25 0 0.5 0.4 0.3 0.1 (b) P(x) 4. 5. 0.2 The probability distribution of the number of calls x that arrive at a switchboard during any one-minute period is shown in the following table. Find the mean, variance and standard deviation of the number of calls. x 0 P(x) 0.02 1 2 3 4 5 6 0.05 0.24 0.17 0.23 0.07 0.22 In a lottery, 10000 tickets are sold for RM2 each. One first prize of RM4000, one second prize of RM1000, one third prize of RM500 and ten consolation prizes of RM100 are to be won. What is the expected net earnings of a person who buys one ticket? Interpret your results. 6. Three chocolate bars are selected randomly from a box containing 10 chocolate bars of which 3 are white chocolates. What are the expected number and the standard deviation of white chocolates? 7. When it is sunny, a man walks to work at no cost. When it is cloudy, he takes a bus, which costs RM1.00. When it is raining, he drives his car, at a cost of RM2.00. It is sunny 50% of the time, cloudy 30% of the time, and raining 20% of the time. Find his average cost of a trip to work. 8. The owner of a construction company makes bids on jobs that if awarded, the company will make RM15000 profit in each job. The following probability distribution describes the number of jobs awarded to the company per year. x ( the number of jobs awarded) P(x) 2 3 4 5 6 0.15 0.30 0.30 0.20 0.05 (a) Find the expected number of jobs awarded to the company in a year? (b) Find the variance and standard deviation of the number of jobs awarded per year for the company? (c) What is the expected profit of the company? What is the standard deviation?