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- Q3/A/ Let X be a r.v with p.d.f f(x) = e-lkl - oProve that if M(t) is the MGF of a random variable X, then the MGF of a + bX is e^at M (bt)Let X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x isSuppose that f (x) = e=* for 0 < x Determine the mean and variance of the random variable.The manager of a bakery knows that the number of chocolate cakes he can sell on any given day is a random variable with probability mass functionPx (0) = 1/12Px(1) = 1/12Px(2) = 3/12Px(3) = 4/12Px (4) = 2/12Px(5) = 1/12He also knows that there is a profit of $1.00 on each cake that he sells and a loss (due to spoilage) of $0.40 on each cake that he does not sell. Assuming that each cake can be sold only on the day it is made, how many chocolate cakes should he bake to maximize his expected profit?Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.SEE MORE QUESTIONS