8. Compute the p.g.f. of X when X has the following distribution.(a) When X has a Bernoulli distribution with parameter p.4(b) When X has a binomial distribution with parameters n and p wherene N and 0 < p < 1.(c) When X has a geometric distribution with parameter p with 0 < p < 1.(d) When X has a Poisson distribution with parameter λ > 0.

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Answered: . Compute the p.g.f. of X when X has…

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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Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
Not sure how to answer questions C through E. Suppose that X has a (continuous) uniform distribution between 10 and 20 (a) Compute E (X). (b) Compute Var (X). (c) Compute P (X < 12). (d) Compute P (X > 12). (e) Compute P (X < 8 or X > 22).
3: Find the theoretical frequencies, by using a suitable distribution X 0 1 2 3 4P(X) 21 / 462 140 / 462 210 / 462 84 / 462 7 / 462
4. The pmf of a geometric distribution with parameter p is given by S p(1 – p)" n= 0, 1, 2, ... otherwise. Pn = Assume p> 0. Show that the variance is (1 – p)/p². Hint: Σ Eng" (1 – q)² n=0 for |g| < 1.
9. N has a conditional Poisson distribution, with A-U(0,2). PZE Calculate: a. E(N) b. V(N) Pr(N> 2) C.
Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter ß. Given Y = y, the number of accidents that the individual suffers in years 1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X₁, Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X₂ = k2 … . . Xn = kn) = [° ƒ (y, k1, k2, ... kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson distribution with parameter y.
A1 We mentioned the use of the Poisson distribution for modelling the number of “events” that occur in a two-dimensional region. Assume that when the region R being sampled has area a(R), the number X of plants of a certain species occurring in R has a Poisson distribution with parameter λa(R) (where λ is the expected number of plants per unit area) and that nonoverlapping regions yield independent X’s. Suppose an ecologist selects n nonoverlapping regions R1, . . ., Rnand counts the number of plants of a that species found in each region. Which is the expression of the joint pmf (likelihood)? Which is the maximum likelihood estimator of λ.
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5. Let X₁ and X2 be independent random variables. Suppose that X₁ follows a Poisson distribution with parameter X₁, and that X2 follows a Poisson distribution with parameter λ₂. Let Y = X₁ + X₂. (a) Use the MGF technique to find the distribution of Y. (b) Find E(Y) and Var(Y).
1. Let X follow a binomial distribution. Show that Var[X] 8). 3. Suppose X follows a binomial distribution with parameters n and p. Let Y = n – X. Show that Y is a binomial distribution with parameters n and 1-p. (To help think about it, think of X as the number of successes and Y is the number of failures.)

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