4. Assume that the joint pmf of (X,Y) is 0.20 0.10 0.05 0.25 0.10 Find (1)...
1. Consider a discrete bivariate random variable (X,Y) with the joint pmf given by the table: Y X 1 2 4 1 0 0.1 0.05 2 0.2 0.05 0 4 0.1 0 0.05 8 0.3 0.15 0 Table 0.1: p(, y) a) Find marginal distributions of X and Y, p(x) and pay respectively. b) Find the covariance and the correlation between X and Y.
4.12 The joint pmf of X and Y is P(X = x, Y = y) = ! 2+1 19 , for x = 0,1 and y = 0,1,2,3. Find the marginal distributions of X and Y. Describe their distributions qualitatively. That is, identify their distribu- tions as one of the known distributions you have worked with (e.g., Bernoulli, binomial, Poisson, or uniform).
Table 1 Joint PMF of X and Y in Example 5.1 x=01 X=1 | 1 Fig. 1 shows PXY()PXY( JointPMF ? 2 Fig. 1. Joint PMF of X and Y (Example 5.1). a. b. c. d. Find P(X-0,Y<1). Find the marginal PMFs of X and Y. Find P(Y-1X-0). Are X and Y independent?
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
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]
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete) and Y (continuous), for x = 1, 2, 3, 4 and 0 <y < 2. (a) Determine the marginal pmf of X. (b) Determine the marginal pdf of Y. (c) Compute P(X<2 and Y < 1). (d) Explain why X and Y are dependent without computing Cou(X,Y).
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)
3. Consider the following joint probability mass functiono Py (y) 0 0.10 0.20 Y 1 0.10 0.40 0.10 Pr() a) Find the missing probability? b) Are X and Y independent? Justify your answer. d) Find the correlation between X and Y.
2. Let X and X be two random variables with the following joint PMF Yix 2 0 2 0 0.1 0.05 0.05 0.15 0.1 0.05 0.1 0.05 0.05 0.05 4 0.05 0.05 0.02 0.1 0.03 total 0.2 0.2 0.12 0.3 0.18 total 0.45 0.3 0.25 1 1) Find E[X] and E[Y]. (10 points) 2) What is the covariance of X and Y? (20 points) 3) Are X and Y independent? Explain. (10 points)
271 Exercise 1/1.4. Consider the joint pmf p(x, y) = cxy. 1 sxsy <3. (a) Find the normalizing constant c. (b) Are X and Y independent? Prove your claim. (c) Find the expectations of X, Y, XY.