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- Consider that a pdf of a random variable X is 1 -25x53 fx (x) ={K otherwise and another random variable Y = 2X. Then find (a) value K, (b) E[X], (c) E[Y] and ( d) EΧΥ.6. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.Let X be a continuous random variable with cdf F(x). Show that E(I(X < x)) = F(r) where I is the indicator function (1 if X8Let Q be a continuous random variable with PDF | 6q(1 – q) if 0 < q < 1 fo(q) = otherwise This Q represents the probability of success of a Bernoulli random variable X, i.e., P(X =1|Q = q) = q. Find foix (q|x) for x E {0, 1} and all q.X is a random variable that has the following PDF function: fx(x) = 0.28(x + 2) + 0.38(x + 1) + 0.358(x) + 0.158(x – 1) %3D For y = x2, find: 1- fy) 2- Fy(y) 3- E[y] 4- oLet X be a random variable with CDF x > 1 Fx(x) = 0 < x < 1 %3D x < 0 a. What kind of random variable is X: discrete, continuous, or mixed? b. Find the PDF of X, fx(x). c. Find E(ex).Let X be a point randomly selected from the unit interval [0, 1]. Consider the random variable Y = (1-X)-¹/2 (a) Sketch Y as a function of X. (b) Find and plot the cdf of Y. (c) Derive the pdf of Y. (d) Compute the following probabilities: P(Y > 1), P(3 < Y < 6), P(Y ≤ 10).Let T be a statistic, then T is sufficient statistic for 6 if the conditional pdf f(8,x1, X2,...,X, \ T)depend on the parameter 0 Select one: True False8. Assume that X is a continuous random variable with pdf 1 if -6Theorem 11. Let X be a random variable and let g(x) be a non-negative function. Then for r > 0, Eg (X) P[g(X) > r] < Proof.12. Let the random variable X and Y have joint pdf 4 f(x,y) = (x² + 3y²), 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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