3. Let X be a discrete random variable with probability mass function p(n) = 7 (1)", Find the moment-generating function Mx(t) with the appropriate domain. n = 1,2,3,....
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A: Solution: From the given information, the moment generating function is
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A: we need to find the MLE of .
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Q: Let X be a random variable with E[X]=2, Var(X)= 4. Compute the expectation and variance of 3– 2X .
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Q: (4) Let X be a random variable with p.d.f. 2e-2x 0<x<0 f(x)= , find E(e2x) O.w
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- Q. 4 Let X and Y be two continuous random variables with following joint probability density function SK (x² + y²); 0 Y), (c) P(X +Y > 1) (b)Let W₁ < W₂ < ... < Wn be the order statistics of n independent observations from a U(0, 1) distribution. (a) Find the pdf of W₁ and that of Wn. (b) Use the results of (a) to verify that E (W₁) = 1/(n+1) and E(Wn) = n/(n+1).1
- 3. 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.Let Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)26. Suppose that the random variables X and Y have joint probability density function given by x+y, 07. Let X~ N (0,0²) and {X; : i = 1,2,..., n} be a random sample from X. (a) Formulate the log-likelihood function. (b) Find the ML estimator of o². (c) Derive the variance of the ML estimator of o2, 62. Does the variance of ô2 achieve the CR bound? (d) Derive the asymptotic distribution of √n (-o).Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(23) 0.34 F(29) = =0.38 F(34) 0.42 F(39) 0.47 F(44) = 0.52 F(49) 0.55 F(56) = 0.61 = Determine Pr[29Let Y1, Y2,., Ya be a collection of independent random variables with distribution function y 8 Show that Y converges in probability to a constant, and provide that constant. 1Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON