Problem 2 (20 points) Suppose (Y,X) e R × 10, îl has joint density 1/2e-0.5(y+) T (a) Find the ma...
(a) Show that fY X(y; x) is a valid density function. (b) Find the marginal density of Y as a functon of the CDF (c) Find the marginal density of X. (d) Deduce P[X < 0:2]. (e) Are Y and X independent? Problem 2: Suppose (Y, X) is continuously distributed with joint density function (a) Show that fyx(y, x) is a valid density function (b) Find the marginal density of Y as a functon of the CDF Φ(t)-let φ(z)dz. (c)...
#2. (24 points) Let X and Y have joint density (a) Find the marginal pdf of Y. Use it to find E(Y) (b) Give an integral expression for P(X + Y < 0.75), but do not evaluate. (c) Give an integral expression for E(XY), but do not evaluate. Optional two point bonus problem. In Problem 2 above, is the distribution of Y skewed to the left or skewed to the right? Explain. #1. (28 points) Suppose that X has probability...
Problem 1. Suppose that X and Y are jointly contimmous with joint probability density function re-r(1+y), İfx > 0 and y > 0. otherwise. (a) Find the marginal density functions of X and h (b) Calculate the expectation of E(XY). (e) Calculate the expectation E
4. Suppose X and Y has joint density f(x, y) = 2 for () < x <y<1. (a) Find P(Y - X > 2). (b) Find the marginal densities of X and Y. (c) Find E(X), E(Y), Var(X), Var(Y), Cov(X,Y)
1. (20 pts) RVs X and Y have joint density function 22 f(x, y) =(0 if O <z<1 and 0<y<2 īf 0 < x < 1 and 0 < y < 2 otherwise (a) Find E(X), V(X), E(Y), and V(Y). (b) Find the covariance cov(X,Y) and the associated correlation ρ (c) Find the marginal densities fx and fy. (Be sure to say where they're nonzero.) (d) Find E(X | Y = 1.5). (e) Are X and Y independent? Give two...
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y...
6. (10 points) Suppose X and Y are not independent, and are given by joint density function fx,y(a, b) = (x + 2y)1ce(0,1,5(0,1). What is the density of X+Y?
4. (20 points) Suppose the joint distribution of X and Y is: fxy(x, y) 1 0 1 2 3 0.04 0.06 0.01 0.00 0.13 0.13 0.02 0.12 0.04 0.06 0.00 0.11 0.07 0.10 0.06 (a) (4 points) Find the marginal distributions of X and Y. (b) (4 points) Given X = 3, what is the probability that random variable Y is at most 2?. (c) (4 points) Are random variables X and Y independent? Why or why not? (d) (4...
1) Suppose that three random variables, X, Y, and Z have a continuous joint probability density function f(x, y. z) elsewhere a) Determine the value of the constant b) Find the marginal joint p. d. fof X and Y, namely f(x, y) (3 Points) c) Using part b), compute the conditional probability of Z given X and Y. That is, find f (Z I x y) d) Using the result from part c), compute P(Z<0.5 x - 3 Points) 2...
)on 4. Suppose X and y are continuous random variables with joint density funstion the unit square [0, 1] x [0, 1]. (a) Let F(r,y) be the joint CDF. Compute F(1/2, 1/2). Compute F(z,y). (b) Compute the marginal densities for X and Y (c) Are X and Y independent? (d) Compute E(X), E(Y), Cov(X,y)