1. [6 points] Suppose that Xi is a normally distributed random variable with ECX
ID: 3072373 • Letter: 1
Question
1. [6 points] Suppose that Xi is a normally distributed random variable with ECXi) 2, vCx)-0.25 and X2 is also normal with E(X) 3 ,V(X) -0.5 Assume that both random variables are independently distributed and let Y 4X1 + 3X2-1, Compute (a) E(Y) (b) V(Y) (c) P(Y> 7.10) [4 points] The lifetime of an insulating material tested (accelerated testing) at 30k V is known to have a mean of 54 hours and standard deviation of 5 hours. A reliability engineer randomly selects n -144 specimens and determines their lifetimes. Approximate the probability of the sample mean exceeding the population mean by 1.5 standard deviations. 2.Explanation / Answer
Solution
a. E(Y) = E(4X1 +* 3X2 - 1) = 4E(X1)+3E(X2) - 1 = 4*2 - 3*3 - 1 = -2
b. V(Y) = V(4X1 + 3X2 - 1) = 16V(X1) + 9 V(X2) - 0
as independent so covariance = 0
V(Y) = 16*0.25 + 9*0.5 = 8.5
c. P(Y>7.1) = P(Z> (7.1+2)/8.50.5 ) = 1 - P(Z<3.12) = 0.0009
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