Question 1: Conditional of Poisson random variabless is Multinomial Lct X1,.... X% be independent random variables...
2. Let X1 and X2 be independent Poisson random variables with parameters λ1 and A2. Show that for every n 21, the conditional distribution of X1, given Xi X2n, is binomial, and find the parameters of this binomial distribution
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find the distribution of x, + x, (b) Compute the conditional distribution of Xi given that X + X.-n
a) If X1 and X2 are independent random variables and X1 tollows the Nor nalLA σ1 X, +X2 follow? di tri t on and X to ows the Nonna μα 2 distribution, ne ha distribution do b) IfX1 , X2 . X, , arendependent random variables and each Xk follows the NormalA 에 ds rbutio. then what distribution does follow? , n L.6) Generating functions for sums of independent random variables a) If X and X are independent random variables,...
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b) Calculate the conditional distribution of Xi given Y-n for n 2 0. Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b)...
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2. (a) Find the conditional distribution of (X1,X2) given T = 7. (b) For θ = 1 and θ = 2, respectively, calculate all probabilities in the above conditional distribution and present the two conditional distributions numerically.
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
Let X1, · · · , X20 be independent Poisson random variables with mean (print please) 1. (i) Use the pmf of Poisson distribution to compute P(X1 + · · · + X20 > 15). (ii) Use the Markov inequality to obtain a bound on P(X1 + · · · + X20 > 15). (iii) Use the central limit theorem to approximate P(X1 + · · · + X20 > 15).
6. Conditioning independent Poisson variables on their sum. Let N, be independent Poisson variables with parameters di. Think of the N, as the number of points of a Poissson scatter in disjoint parts of the plane with areas li, where the mean intensity is one point per unit area. a) What is the conditional joint distribution of (N1, ... , Nm) given Ni+...+Nm = n? (Hint: See Exercise 5 for a special case.) b) Suppose now that N has Poisson(2)...
7. Suppose that Xi,..., Xk are independent random variables, and X, ~ Exp(B) for i = 1, . . . , k. Let Y = min(X1 , . . . , Xk). Show that Y ~ Exp(Σ-1 β).