2. 6. (20points). Let X and Y have the joint pmf ? 1,2,3 , zero elsewhere....
2. Let Px(x) = 1, X = 1,2,3, 4, 5, zero elsewhere, be the pmf of X. Find P(X = 1 or 2), P(3 < X < ), and P(1 < X < 2).
2. Let X1 and X2 have the joint pmf p(11, 12) = 232, 21 = 1,2,3 and x2 = 1,2,3, zero elsewhere. Find the joint pmf of Y1 = X1 X2 and Y2 = X2, and then find the marginal pmf of Y.
4.2 The Correlation Coefficient 1. Let the random variables X and Y have the joint PMF of the form x + y , x= 1,2, y = 1,2,3. p(x,y) = 21 They satisfy 11 12 Mx = 16 of = 12 of = 212 2 My = 27 Find the covariance Cov(X,Y) and the correlation coefficient p. Are X and Y independent or dependent?
Suppose that X and Y have joint pmf px yx,y) fxy-/39 for x 1,2 and y 2,3 0elsewhere). a) Determine the marginal pmfs px(x) and py(y) b) Determine the conditional pmf of px(xly). c) Are X and Y independent? Give a clear determination using probability formulas.
3. Let (X, Y) be a bivariate random variable with joint pmf given by x= 1,2,3, y = 0,1,2,3, ... ,00 f(x, y) 12 0 e.w. (a) Show that f(x, y) is a valid joint pmf. (b) Find fa(x) (i.e. the marginal pmf of X). (c) Find fy(y) (i.e. the marginal pmf of Y). (d) Find P [Y X]
Let the joint pmf of X and Y be p(x, у) схуг, x-1,2,3, y-12. a) Find constant c that makes p(x, y) a valid joint pmf. c) Are X and Y independent? Justify d) Find P(X+Y> 3) and PCIX-YI # 1)
The following table presents the joint probability mass function pmf of variables X and Y 0 2 0.14 0.06 0.21 2 0.09 0.35 0.15 (a) Compute the probability that P(X +Y 3 2) (b) Compute the expected value of the function (X, Y)3 (c) Compute the marginal probability distributions of X and )Y (d) Compute the variances of X and Y (e) Compute the covariance and correlation of X and Y. (f) Are X and Y statistically independent? Clearly prove...
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
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).
We have the following joint PMF of X and Y: Pxy (x,y) ſa(x+3y) x =1,2,3; y =1,2 0 otherwise Find: 1. the value of a 2. the marginal PMFs of X and Y 3. if X and Y are independent