2. (30 pts) Let X and Y be independent rv's with pmf Pois(11) and Pois(2), respectively....
2. (30 pts) Let X and Y be independent rv's with pmf Pois(41) and Pois(12), respectively. (a) Find the distribution of Z = X+Y. (b) Find the distribution of X|X +Y.. (c) If X ~ Pois(11) and Y|X = x ~ Bin(x,p). Find the distribution of Y.
2. (30 pts) Let X and Y be independent rv's with pmf Pois(41) and Pois(12), respectively. (a) Find the distribution of Z = X +Y. (b) Find the distribution of X X +Y. (c) If X ~ Pois(11) and Y|X = x ~ Bin(x,p). Find the distribution of Y.
Let X and Y be independent rv's with pmf Pois(11) and Pois(12), respectively. (a) Find the distribution of Z = X+Y. (b) Find the distribution of X X +Y. (c) If X Pois(11) and Y|X = r ~ Bin(x,p). Find the distribution of Y.
Let X and Y be independent rv’s with pmf Pois(λ1) and Pois(λ2),
respectively.
(a) Find the distribution of Z = X + Y .
(b) Find the distribution of X|X + Y .
(c)If X∼Pois(λ1) and Y|X=x∼Bin(x,p). Find the distribution of
Y.
Let X and Y be independent rv's with pmf Pois(11) and Pois(12), respectively. (a) Find the distribution of Z= X+Y. (b) Find the distribution of X|X +Y. (c) If X ~ Pois(11) and Y|X = x ~ Bin(x,p)....
Let X, Y Geometric(p) be independent, and let Z a. Find the range of 2. b. Find the PMF of Z c. Find EZ.
Let X, Y Geometric(p) be independent, and let Z a. Find the range of 2. b. Find the PMF of Z c. Find EZ.
13. Let X and Y be rv's whose joint PMF is given by: Y=1 2 3 X=0 0.2 0.1 0 1 0.1 0.3 0. 2 . 0 0 0.3 Compute the covariance and correlation matrix of the random vector (X,Y).
Let X and Y be independent exponentially distribution
random variables with rate α and β respectively. Find P (X > Y
).
Question 13: Let X and Y be independent exponentially distribution random variables with rate a and B respectively. Find P(X> Y).
(f) Find the conditional pmf of X given Z. Identify this
conditional distribution as a distribution known in class, and
give
the explicit parameters for the known distribution.
(g) Find the conditional expectation of X given Z.
2. (Lec 13 &15 & 16 pairs of discrete R.V., conditional pmf and conditional moments, 17 pts) We are studying the flow of packets at a switch, which receives packets from two transmission paths, during a given period of time. Let X and...
Let the joint pmf of X and Y be defined by x+y 32 x 1,2, y,2,3,4 (a) Find fx(x), the marginal pmf of X. b) Find fyv), the marginal pmf of Y (c) Find P(XsY. (d) Find P(Y 2x). (e) Find P(X+ Y 3) (f) Find PX s3-Y) (g) Are Xand Y independent or dependent?Why or why not? (h) Find the means and the variances of X and Y
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...