Prove a formula for the moment-generating function (mgf) of Y in terms of the mgf's of X₁ and N. For any random variable Z, the mgf is defined as oz(s) = E[e³Z], where s € IR and whenever the expectation is defined. Let X₁, X2,..., be a sequence of independent and identically distributed random variables, each with a mean value ux and variance o, both of which are assumed to be finite. Let N be a discrete random variable independent of X₁, X2, ..., and assuming values in the set {0, 1, 2,...}, with mean value μN and variance ok (both are finite). Form the random-compound sum Y = 1 Xk, with the convention Y = 0 whenever N = 0.

Prove a formula for the moment-generating function (mgf) of Y in terms of the mgf's of X₁ and N. For any random variable Z, the mgf is defined as oz(s) = E[e³Z], where s € IR and whenever the expectation is defined. Let X₁, X2,..., be a sequence of independent and identically distributed random variables, each with a mean value ux and variance o, both of which are assumed to be finite. Let N be a discrete random variable independent of X₁, X2, ..., and assuming values in the set {0, 1, 2,...}, with mean value μN and variance ok (both are finite). Form the random-compound sum Y = 1 Xk, with the convention Y = 0 whenever N = 0.

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...

See similar textbooks

Similar questions

Recall that two events A and B are called independent if p(AB) = p(A)p(B). Similarly two random variables x and y are called independent if the joint probability function f(x, y) = g(x)h(y). Show that if x and y are independent, then the expectation or average of xy is E(xy) = E(x)E(y) = µxµy.
Let X be a random variable with pdf f(x) = 0.2 €-0.2x 0 if x > 0, otherwise. (a) Find the moment generating function of X, remembering to state the range for which it holds. (b) Hence find the expectation and variance of X.
we are given independent random variables X and Y distrubuted: X ∼ poisson(θ) , Y ∼ poisson(2θ), and observations x = 3 and y = 5. Show that the expression for the log-likehood function is given by: l(θ)=[5ln(2)−ln(3!)−ln(5!)]+8lnθ−3θ. make a sketch of l(θ) for θ ∈ [0, 10]. for which value of θ does l(θ) reach its maximum?
Show that al(w) is a random variable, where EEF and a is a constant. Calculate E[X] using the basic definition of an expectation as done in class.
Suppose the random variable X has pdf K exp £x(u) = 2√/27 x ( - (a - 2²). - where A is a constant to be determined. Determine K, the CDF Fy, and the mean E[X]. 0 ≤ ≤ 4, otherwise.
Suppose that the random variables Y and X only take the values 0 and 1, and have the following joint probability distribution: Y = 0 Y = 1 X = 0 .4 .1 X = 1 .3 .2 Find E[Y | X], E [Y² | X] and Var[Y | X] for X = 0 and X 1. You may use the fact that Var[Y | X] = E[Y² | X] – E[Y | X]² without a proof.
Let X, Y, and Z be random variables, and let Cov(⋅,⋅) denote the covariance operator as usual. Suppose that the variance of X is 0.7, Cov(X,Y) = 0.4, Cov(X,Z) = 1.2, and Cov(Y,Z) = 0.8. Find each of the following to two decimal places. (a) Cov(3Y, 3X) (b) Cov(3Y + 3, 3X + 8Z)
Let Y1 and Y2 be independent Poisson random variable with means , and 22, respectively. Using a moment generating function technique, find the probability density function of U=Y1+Y2. Hence, find the conditional probability function of Y1 given Y1+ Y2 = m. 4.
For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 1.
Assume the lifetime of electronic systems T (based on months) has Exponential distribution with λ=0.01. We select a random sample of 5 systems (random lifetime of these systems are T1, T2, T3, T4, and T5). Assume, S= T1+T2+ T3+ T4+ T5. a. Find E[S] . b. Find Standard Deviation of S (σSσS). c. Find median of T1.
Let X and Y be standard normal random variables. If S = px + √√1-p²Y, derive the distribution of S by using only the moment generating function (mgf) method. (Hint: If Z follows the normal, N(µ,o²) distribution, then the mgf of Z is M₂(t) = + 0²1²) e
Let Y denote a random variable with the following cumulative distribution function. if y 1. F(y) : In(4) a. Find the PDF, ƒ(y). b. Evaluate P(Y = 0.5). c. Evaluate P(Y 0.4). d. Evaluate P(Y < 1).