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- If X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).Let X ~ U[0,1] and Y = -βln(1-X). What is the distribution of Y? Justify.suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)
- Suppose that Y1, . . . , Yn is a random sample from a population whose density function isFor c in R, calculate E[e^cX] where X is exponential distribution.Let X be a random variable with exponential distribution having lambda=6 and let Y = 3X. a. Find P(X>0.25) b. Find the mgf of Y c. Find the pdf of Y
- 1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.For the continuous probability function f(x ) = kx^2e^-x when 0≤x≤1. Find (a)k (b)mean (c)varianceIf the probability density of X is given by f(x) =2x−3 for x > 10 elsewherecheck whether its mean and its variance exist.