Exam 3 Fall 2003 2.(20 points) The lengths X and Y of two sides of a...
1. Let X and Y be continuous random variables with joint pr ability density function 6e2re5y İfy < 0 and x < otherwise. y, fx,y (z,y) 0 (a) [3 points] Show that the marginal density function of Y is given by 3es if y 0, 0 otherwise. fy (y) = (b) |3 poin s apute the marginal density function of X (c) [3 points] Show that E(X)Y = y) =-y-1, for y 0 (d) 13 points] Compute E(X) using the...
a. Given the joint probability den- sity function fxy(x, y) as, Skxy, (x, y) e shaded area Jxy(, 9) = 10 otherwise Find [i] k [ii] fx(x) [iii] fy(y) Are X and Y independent? b. Given the joint probability density function fxy(x, y) as, fxy(x, y) = { 0 kxy, (x, y) E shaded area otherwise Find [i] k [ii] fx(x) [iii] fy(y) Are X and Y independent? 2 1
I . (20%) Random variable X has the probability density function as ; Random variable Y 2X+1 0 otherwise a) Determine A b) Determine the Probability Distribution Function F, (x) c) Determine E(X) and ơx d) Determine the probability density function fy(y) and E(Y)
4. Suppose that a two-dimensional random vector (X, Y) has a joint probability density function as 0.48y(2-x), 0 1,0 x y x f(x,y)- 0, otherwise Find two possible marginal probability functions fx(x) and fy(y) of X and Y, respectively. 4. Suppose that a two-dimensional random vector (X, Y) has a joint probability density function as 0.48y(2-x), 0 1,0 x y x f(x,y)- 0, otherwise Find two possible marginal probability functions fx(x) and fy(y) of X and Y, respectively.
between 0 and 4, x-UlO,4]. Another random variable, Y, is given Q1) Random variable as a function of g(x), Y X has uniform distribution g(x) where g(x)- 3-х, 2 x < 3. 0, otherwise. For parts a, b, and c, plotting the function y g(x) can be very useful. a-What is P(Y 0) [4 points] b-What is P(Y 1) 13 points] c-Derive and plot the cumulative distribution function (CDF) of Y, Frv). [10 points) d-What is probability distribution of Y,...
between 0 and 4, x-UlO,4]. Another random variable, Y, is given Q1) Random variable as a function of g(x), Y X has uniform distribution g(x) where g(x)- 3-х, 2 x < 3. 0, otherwise. For parts a, b, and c, plotting the function y g(x) can be very useful. a-What is P(Y 0) [4 points] b-What is P(Y 1) 13 points] c-Derive and plot the cumulative distribution function (CDF) of Y, Frv). [10 points) d-What is probability distribution of Y,...
4. Two random variables X and Y have the following joint probability density function (PDF) Skx 0<x<y<1, fxy(x, y) = 10 otherwise. (a) [2 points) Determine the constant k. (b) (4 points) Find the marginal PDFs fx(2) and fy(y). Are X and Y independent? (c) [4 points) Find the expected values E[X] and EY). (d) [6 points) Find the variances Var[X] and Var[Y]. (e) [4 points) What is the covariance between X and Y?
(3x, The joint density function of X and Y is given by 0 Sy sxs1 f(x, y) = 0, otherwise. a) Use the distribution function technique to find the distribution function of W = X-Y. For 50% of the points, you may use the transformation technique, which is longer. >) Find the probability density function of W. Find the expected value E(W). )
Exercise 10.33. Let (X,Y) be uniformly distributed on the triangleD with vertices (1,0), (2,0) and (0,1), as in Example 10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You might first deduce the answer from Figure 10.2 and then check your intuition with calculation. (b) Verify the averaging identity for P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:∞ −∞ P(X ≤ 1 2|Y =y)fY(y)dy. Example 10.19. Let (X, Y) be uniformly distributed on the...