3. The pair of random variables X and Y is uniformly distributed on the interior of...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2). 3. The pair of random variables X and Y is uniformly distributed on the interior...
(3) A pair of random variables (X, Y) is distributed uniformly on the triangle with vertices (0,0), (2,0) and (0. Find EX, EY, Cov(X,Y), E(max{X,Y)), P(X> Y), P(X 2 Y)
2. A random vector (X,Y) is uniformly distributed in a triangle with vertices ABC, having coordinates A(0,0), B(2.0), C(0,1). The joint density f(x,y) is given by the formula 141) - f(x,y) = { s 15 inside the triangle outside
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
Exercise 6.34. Let (X,Y) be a uniformly distributed random point on the quadri- lateral D with vertices (0,0), (2,0), (1,1) and (0,1). (a) Find the joint density function of (X,Y) and the marginal density functions of X and Y. (b) Find E[X] and E[Y]. (c) Are X and Y independent?
5. Let (X, Y) be a uniformly distributed random point on the quadrilateral D with vertices (0,0), (2,0),(1,1), (0,1) Uniformly distributed means that the joint probability density function of X and Y is a constant on D (equal to 1/area(D)). (a) Do you think Cov(X, Y) is positive, negative, or zero? Can you answer this without doing any calculations? (b) Compute Cov(X, Y) and pxyCorr(X, Y)
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Let X and Y have a density, f, that is uniform over the interior of the triangle with vertices at (0,0), (2,0), and (1,2). Find the conditional expectation of Y given X.
4. Let X and Y be independent standard normal random variables. The pair (X,Y) can be described in polar coordinates in terms of random variables R 2 0 and 0 e [0,27], so that X = R cos θ, Y = R sin θ. (a) (10 points) Show that θ is uniformly distributed in [0,2 and that R and 0 are independent. (b) (IO points) Show that R2 has an exponential distribution with parameter 1/2. , that R has the...