Exercise 5.6. Suppose that X is a discrete random variable with probability mass function P(X = -1) = 1/7, P(X= 0) = 1/14, P(X= 2) = 3/14, P(X= 4) = 4/7. Compute the probability mass function of (X-1)².
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- Let X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.Suppose Y1, Y2, Y3, and Y4 are mutually independent and identically distributed exponential random variables with mean 1. a) Find the moment generating function of the sum T = Y+Y2+Y3+Y4. What is the distribution of T? b) Find the pdf of the maximum order statistic Y(4).