Math 370X - Quiz 9Saumil PadhyaNovember 7, 2016Name Net ID1. An actuary models the lifetime of a device using the random variable Y = 10X0.8where Xis an exponential random variable with mean 1 per year. Determine the probability densityfunction f(y), for y>0, of the random variable Y.(A) 10y0.8e−y−0.2(B) 8y−0.2e−10y0.8(C) 8y−0.2e−(0.1y )1.25(D) (0.1y)1.25e−0.125(0.1y )0.25(E) 0.125(0.1y)0.25e−(0.1y )1.252.The random variable X has an exponential distribution with a mean of 1. The randomvariable Y is defined to be Y = 2lnX. Find fY(y), the pdf of Y.13.The total claim amount for a health insurance policy follows a distribution with densityfunctionf(x) =11000e−x/1000, x > 0The premium for the policy is set at 100 over the expected total claim amount. If 100 policies aresold, what is the approximate probability that the insurance company will have claims exceedingthe premiums collected?Hint: “Approximate” probability means you should use normal approximation.(A) 0.001(B) 0.159(C) 0.333(D) 0.407(E) 0.4604.X and Y are independent random variables with common moment generating functionM(t) = et2/2. Let W = X + Y and Z = Y - X.Determine the joint moment generating function M(t1, t2) of W and Z.(A) e2t21+2t22(B) e(t1−t2)2(C) e(t1+t2)2(D) e2t1t2(E)
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