5. Suppose that Xn ~ Binomial(n,츰) for n 1.2, and X ~ Poisson(λ). Prove that Xn converges in distribution to X by using the moment generating functions for Xn and X
4. Given a Poisson process X(t), t > 0, of rate λ > 0, let us fix a time, say t-2, and let us consider the first point of X to occur after time 2. Call this time W, so that W mint 2 X() X(2) Show that the random variable W - 2 has the exponential distribution with parameter A. Hint: Begin by computing PrW -2>] for 4. Given a Poisson process X(t), t > 0, of rate λ...
Let A > 0 be fixed and for each n - 1,2,3.., let Xn be a Binomial Random variable with parameters n, and pn -^. (i.e The number of trials is n and thıe success probability is pn --) (a) Write the moment-generating-function, Mx (t of X,. (You do not have to 72 derive it from scratch. You may use the general formula for the mgf of a binomial variable as provided in the appendix of the text). (b) Show...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
fx (z)='0 otherwise Let Xa)<...<Xn) be the order statistics. Show that Xa)/X(n) and X(n) are independent random variables.
(4) Suppose Λ ~ Exponential(7) and X ~ Poisson(A). Use generating functions to show that X + 1 ~ Geometric(p) and determine p in terms of γ. (4) Suppose Λ ~ Exponential(7) and X ~ Poisson(A). Use generating functions to show that X + 1 ~ Geometric(p) and determine p in terms of γ.
Need help plz Let X be exponential with parameter λ. a. What are Fx(xXxo) and fr(alX <xo)? b. What is the conditional mean E[XLX <Xo]? 7.6 is exponential with parameter 1, what X What are the density and distribution of Y What are the 7.9 lf θ ~U(0, 2n): a. What are the density and distribution function of Y= cos(θ)? b. What are the mean and variance of Y? th a Matlab one- 7.11 e.g., u For X exponential with...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
6. Let X have exponential density f(x) = le-Az if x > 0, f(x) = 0 otherwise (>0). Compute the moment-generating function of X.
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .