12. Let X the exponential random variable with parameter å.(a) Show, by deriving from the definition, that the moment-generating function of X isMx (t) =-入-t(b) Using an appropriate property, derive from the formula mentioned in (a) to find the1general formula of E(X*) for k=1,2,.Then verify that E(X)=-and V(X)=え

Answered: Image /qna-images/answer/6941f0a0-bc62-4fab-ab26-e58932ca6958.jpg

Answered: 2. Let X the exponential random…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Q3. (a) Define a function g(x) of a random [ 1; if x2X, 0; otherwise variables by g(x)% = Where x, is a real number -o< X, <∞ Show that E [g (X)] = 1-F, (x,)
10 If 0₁ <0₂, derive the moment-generating function of a random variable that has a uniform distribution on the interval (0₁, 0₂). Suppose U has a uniform distribution on the interval (0₁, 0₂). Then, the moment generating function of U is derived as follows. mu(t)= E = = ·0₂ 7⁰2 du
7. Suppose that the random variable X have the pdf f (x) = e-a*/2, x > 0 and 0 otherwise. 2T (a) Find E(X) and Var(X). (b) Find the transformation g(X)= Y and values of a and B such that Y ~ I(a, B).
Suppose P(X = 2) = 1/2, P(X = 5) = 1/3 and P(X = 7) = 1/6(a) Compute the probability generating function rX(t)(b) Verify that r'X(0) = P(X = 1) and rX”(0) = 2P(X = 2)(c) Compute the moment generating function of X, mX(s)(d) Verify that m'X(0) = E(X) and mX”(0) = E(X2)
69. Let X~ Geom (p) with pmf f(x) = P(X = x) = p(1 − p)ª−¹I{1,2,...} (x). (a) Use the definition of the moment generating function of a discrete random variable to find the moment generating function mx (t) for X. (b) Use the mgf for X to derive the formula for the mean of X, μ = E(X) = m'x (0).
iid Let X1,· where f(xi, Find the log-likelihood .function ., Xn * f(x;, 0), • 0) = Q³e-o°zi O 93n e-0°x; 3nln (θ) e- ' Σ 3nln(0) – 0 C–1 *; i=1 3nln(0) – 0šn xi O None
7. Suppose that X is a continuous random variable with pdf Se-(x+2), 2), if-2 < x < 0 f(x)= = lo, elsewhere a) Find the moment generating function of X. b) Use the MGF of (a) to find E(X) and E(X²). 8.
True false d,e,f
1. Fix r € N, and suppose that Y₁,..., Yn are independent and identically distributed NegBin(r, ) random variables, each with probability mass function given by ¡v (u | 0) – (1 + r − ¹) (1 - fy y (1 - 0) ³0⁰ for y ¤ N and unknown 0 € (0, 1). Here the number of successes r in the negative binomial is known.
2) For a random variable X, its kth-order moment is defined as E(X*). Given the p.d.f. fx(x) of X, its Fourier transform Fx(u) = S fx(x)e¬j2muz dx is also called moment generation function. a) Determine an expression of E(X*) in terms of Fx(u). Hint: consider the kth-order derivative of Fx (u) with respect to u and examine its relationship to the definition of E(X*). b) If X is N(0, o²), we know that Fx (u) = exp (-}(2u)°o²). Apply the above idea to determine E(X3) and E(Xª) in terms of ơ².
Let Y be a random variable with the pdf 1 (y/3) exp(-y/6), 0

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