Population Standard Deviation of Discrete Random Variable一个离散型随机变量的总体标准偏差.pptVIP

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Population Standard Deviation of Discrete Random Variable一个离散型随机变量的总体标准偏差.ppt

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Population Standard Deviation of Discrete Random Variable一个离散型随机变量的总体标准偏差

Example: meal plans (cont.) Jumbo plan for football players Y=daily amount spent E(Y) = $24.75, SD(Y) = $9.50 Amount by which football player’s spending exceeds regular student spending is Y-X E(Y-X)=E(Y)–E(X)=$24.75-$13.50=$11.25 SD(Y ? X) ≠ SD(Y) ? SD(X) = $9.50 ? $7=$2.50 For random variables, X+X≠2X Let X be the annual payout on a life insurance policy. From mortality tables E(X)=$200 and SD(X)=$3,867. If the payout amounts are doubled, what are the new expected value and standard deviation? Double payout is 2X. E(2X)=2E(X)=2*$200=$400 SD(2X)=2SD(X)=2*$3,867=$7,734 Suppose insurance policies are sold to 2 people. The annual payouts are X1 and X2. Assume the 2 people behave independently. What are the expected value and standard deviation of the total payout? E(X1 + X2)=E(X1) + E(X2) = $200 + $200 = $400 The risk to the insurance co. when doubling the payout (2X) is not the same as the risk when selling policies to 2 people. * * * QTM1310/ Sharpe * QTM1310/ Sharpe * 4.2 (cont.) Standard Deviation of a Discrete Random Variable First center (expected value) Now - spread 4.2 (cont.) Standard Deviation of a Discrete Random Variable Measures how “spread out” the random variable is Summarizing data and probability Data Histogram measure of the center: sample mean x measure of spread: sample standard deviation s Random variable Probability Histogram measure of the center: population mean m measure of spread: population standard deviation s Example x 0 100 p(x) 1/2 1/2 E(x) = 0(1/2) + 100(1/2) = 50 y 49 51 p(y) 1/2 1/2 E(y) = 49(1/2) + 51(1/2) = 50 The deviations of the outcomes from the mean of the probability distribution xi - μ ?2 (sigma squared) is the variance of the probability distribution Variation Variation Variance of discrete random variable X Probability Great 0.20 Good 0.40 OK 0.25 Economic Scenario Profit ($ Millions) 5 1 -4 Lousy 0.15 10 P(X=x4) X x1 x2 x3 x4 P P(X=x1) P(X=x2) P(X=x3) P. 207, Handout 4.1, P. 4 Example ?2 = (x