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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Given the moment-generating function MX(t) =e3t+8t2 , find the moment-generating function of the ran-dom variable Z = 1 4 (X − 3), and use it to determine the mean and the variance of Z.If X1 and X2 constitute a random sample of size n = 2from an exponential population, find the efficiency of 2Y1relative to X, where Y1 is the first order statistic and 2Y1and X are both unbiased estimators of the parameter
- Given the moment generating function MX(t) = e 3t+8t2 , find the moment generating function of the random variable Z = 4(X − 3), and use it to determine the mean and the variance of Z.Find the variance by calculating the first two moments of the random variable X = (- 1 / λ) ln (1-U), where U ~ U (0,1) and λ> 0.Let X be a random variable with mean μ and variance _2. Show that E[(X − b)2], as a function of b, is minimized when b = μ.
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- If X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal placesLetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;The probability function of the random variable X is defined as f(x)=cx²(1-x)² for 0<x<1, otherwise f(x)= 0. Calculate the constant c , the expected value and the variance.