4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(...
4. The random variables X and Y have joint probability density function fx.y(r, y) given by: else (a) Find c (b) Find fx (r) and fr (u), the marginal probability density functions of X and Y, respectively (c) Find fxjy (rly), the conditional probability density function of X given Y. For your limits (which you should not forget!), put y between constant bounds and then give the limits for r in terms of y. (d) Are X and Y independent?...
Q.4 (22') Suppose the joint probability density function of X and Y is fx,y(x, y) = { „) - k(2 - x + y)x 0 sxs 1,0 sys1 o otherwise (a) (7”) Show that the value of constant k = 12 (b) (7') Find the marginal density function of X, i.e., fx(x). (c) (8') Find the conditional probability density of X given Y=y, i.e., fxy(xly). 11
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.
3) The joint density function of X and Ý is given by fx,y) = xex(ri)〉0, y >0 a. By just looking at f(x.y), say ifX and Y are independent or not. Explain. b. Find the conditional density of X, given Y-y. In other words, fy(xly). c. Find the conditional density of Ý, given X=x.
The joint probability density function of random variables X and Y is given by f(x,y) ={10xy^2 0≤x≤y≤1,0 otherwise. (a) Compute the conditional probability fX|Y(x|y). (b) Compute E(Y) and P(Y >1/2). (c) Let W=X/Y. Compute the density function of W. (d) Are X and Y independent? Justify briefly.
Random variables X and Y have the following joint probability density function, fx,y(x, y) = {c)[4] < 15.36, 1y| < 15.367 1.36} 0, 0.w. where cis a constant. Calculate P(Y – X| < 8.41).
Suppose X and Y have joint probability density function fX,Y(x,y)=70e?3x?7y for 0<x<y; and fX,Y(x,y)=0 otherwise. Find E(X). (You may either use the joint density given here,
[1] The joint probability density function of two continuous random variables X and Y is fx,x(x, y) = {6. sc, 0 <y s 2.y = x < 4-y otherwise Find the value of c and the correlation of X and 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.