Problem 4. Suppose the moment generating function for a random variable X is Mx(t) = 0.2 + 0.3e'+0.5e",-o0
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- 3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.4. The lifetime in hours of an electronic gadget is a random variable having a probability density function of x2 0. Compute the expected lifetime of such a gadget. f(x) = xe-2x1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).
- (24) Let X be a random variable with p.d.f. Ze-2 07. If a random variable has the probability density function -1≤x≤3 otherwise f(x)=k(x²-1) =02. (5 ts) Suppose X is a discrete random variable with probability mass function (pmf) f(x) = c(x - 1)2 for r= -4,-2,2,4, | where c is a constant. (a) Find c. (b) Find P(-2 < X < 5).1 (20') The discrete type random variable X has PMF Px(1)= { c(5- ) for a= 1,2, 3, 4 0, otherwise. (a) Determine the value of c. (b) Find E[X] and Var[X]. (c) Find the probability P[X > 2].5. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³3. Let X1, X2, .., X100 be a random sample of size n = 100 from the distribution with p.m.f. f(x,) = (0.1)* (0.9)'-, x, = 0,1. 100 (a) Let Y= X, . Find the approximate probability P(12 < Y< 15), using the Poisson i=1 approximation. (b) Find the approximate probability P(12 < Y< 15), using the normal approximation.3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for t5. Suppose that the random variable X follows an exponential distribution with probability density function S(z) = de-A";0 0). Define a new random variable Y = x. (a) Show that the cumulative density function of Y is given by Fy(y) = {-exp(-Ay"; and hence, or otherwise, find the probability density function of Y. (b) Find an expression for the maximum likelihood estimator of the parameter A, using a sample Y1, 92, , Yn, from the distribution of Y. v20Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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