®, X and Y are two random variables denoting the proportions of time during which employees...
③. X and Y are two random variables denoting the proportions of time during which employees I and II work on a weekday respectively. The joint density of X and Y is f(r,g)-elsewhere (a) Find P(X 1's 흘). Hint: Definition of conditional probability: (b) Find P(J2 |Y = 1). P(A|B) = P(An B)/P(B).
7. Two random variables X and Y have joint probability density function s(x, y) = $(1 – xy), 0<x< l; 0<y<l. The marginal pdfs for X and Y are respectively S(x) = {(2-x) 0<x< 1; s()= (2-y) 0<y<l. Determine the conditional expectation E(Y|X = x) and hence determine E(Y) [7] (ii) [3] Verify your answer to part (i) by calculating the value of E(Y) directly from the marginal pdf for Y. [Total 10]
Consider two random variables X and Y be the proportion of time that a person travels to work by bus and MTR respectively (there are other modes as well). Moreover, X and Y has the following joint distribution f(x,y) 24xy, 0 x f(x,y)-0 otherwise 1,0 y 1, x + y 1 () Find the marginal distributions g(x) and h(y) respectively. (ii) Find the conditional density function fxly) (ii Ifit is known that a person has 0.75 chance of using bus,...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
1. Let X and Y be random variables with joint probability density function flora)-S 1 (2 - xy) for 0 < x < 1, and 0 <y <1 elsewhere Find the conditional probability P(x > ]\Y < ).
The joint density of random variables X and Y is given to be f(x,y) =xy^2 for 0≤x≤y≤1 and is 0 elsewhere. (a) Compute the marginal densities for X and for Y respectively. (b) Compute the expected valueE(XY). (c) Define a new random variable W=Y/X. Compute the probability P(W > t) for anyt >1. Also find the probability P(W <1/2) ?
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
55. Let X and Y be jointly continuous random variables with joint density function fx.y(x,y) be-3y -a < x < 2a, 0) < y < 00, otherwise. Assume that E[XY] = 1/6. (a) Find a and b such that fx,y is a valid joint pdf. You may want to use the fact that du = 1. u 6. и е (b) Find the conditional pdf of X given Y = y where 0 <y < . (c) Find Cov(X,Y). (d)...
The joint probability density function for continuous random variables X and Y is given below. f (x) = x + y, 0 < x < 1, 0 < y < 1 if; 0, degilse. (a) Show that this is a joint density function. (b) Find the marginal density of X . (c) Find the marginal density of Y . (d) Given Y = y find the conditional density of X . (e) P ( 1/2 < X < 1|Y =...