Question: If n independent random variables Xi have normal distributions with
If n independent random variables Xi have normal distributions with the means µi and the standard deviations σi, find the moment-generating function of their sum and identify the corresponding distribution, its mean, and its variance.
Answer to relevant QuestionsProve the following generalization of Theorem 7.3: If X1, X2, . . ., and Xn are independent random variables and Y = a1X1 + a2X2 + · · · + anXn, then Where MXi(t) is the value of the moment-generating function of Xi at t. ...If X1 and X2 are independent random variables having exponential densities with the parameters θ1 and θ2, use the distribution function technique to find the probability density of Y = X1 + X2 when (a) θ1 ≠ θ2; (b) ...Use a computer program to generate 10 “ pseudorandom” numbers having the standard normal distribution. If the number of minutes it takes a service station attendant to balance a tire is a random variable having an exponential distribution with the parameter λ = 5, what are the probabilities that the attendant will take (a) ...Use the corollary of Theorem 4.15 on page 136 to show that if X1, X2, . . . , Xn constitute a random sample from an infinite population, then cov(Xr – , ) = 0 for r = 1, 2, . . . , n. Theorem 4.15 If the random variables ...
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