Previous Quiz - Q.3Lecture 4 ExamplesMath 370X - Lecture 4 ExamplesExpectation and Other Distribution ParametersSaumil PadhyaOctober 3, 2016Previous Quiz - Q.3Let X be a random variable with distribution functionF (x) =0 for x < 0x8for 0 ≤ x < 114+x8for 1 ≤ x < 234+x12for 2 ≤ x < 31 for x ≥ 3Calculate P (1 ≤ X ≤ 2).(A)18(B)38(C)716(D)1324(E)19241Lecture 4 ExamplesExample 1:Let X equal the number of tosses of a fair die until the first “1” appears. Find E[X].Example 2:A fair die is tossed until the first 1 appears. Let x equal the number of tossesrequired, x = 1, 2, 3,. . . You are to receive (.5)xdollars if the first 1 appears on the xthtoss.What is the expected amount that you will receive?Example 3: A continuous random variable X had density functionfX(x) = 1 − |x| if |x| < 1Find the mean and variance of X.Example 4:The pdf of X is f(x) = 5e−5xfor x > 0. Find the mgf of X and use it to find thefirst and second moments of X, and the variance of X.Example 5:The continuous random variable X has pdf f(x) =12· e−|x|for -∞< x <∞. Findthe 87.5thpercentile of the distribution.Example 6:The pdf of X is fX(x) = 2x for 0 < x < 1. Find the mean and cdf of the conditionaldistribution of X given that X ≤12.Example 7:Each of the following is a mgf for a discrete non-negative integer-valued randomvariable. For each random variable, find the mean, variance and the probability P(X ≤ 2).(a)16(1 + 2et+ 3e2t).(b) e2et−2.Example 8:A probability distribution of the claim sizes for an auto insurance policy is given inthe table below:Claim Size Probability20 .1530 .1040 .0550 .2060 .1070 .1080 .30What percentage of the claims are within one standard deviation of the mean claim
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