We have X and Y that are exchangeable in the sense that (X,Y) = (Y,X). That is, their joint pdf f(x,y) satisfies f(x,y) = f(y,x), for all (x,y).
Show that fY-X (y-x) = fY-X -(y-x).
We have X and Y that are exchangeable in the sense that (X,Y) = (Y,X). That...
4.4-2. Let X and Y have the joint pdf f(x, y) r + y, = x + y, (a) Find the marginal pdfs fx(t) and fy (v) and show that f(x,y)关fr (x)fy(y). Thus, X and Y are dependent. (b) Compute (i) μ x, (ii) μ Y. (111) 07, and (iv) 어.
4. Suppose X and Y have the joint pdf f(x,y) = 6x, 0 < x < y < 1, and zero otherwise. (a) Find fx(x). (b) Find fy(y). (c) Find Corr(X,Y). (d) Find fy x(y|x). (e) Find E(Y|X). (f) Find Var(Y). (g) Find Var(E(Y|X)). (h) Find E (Var(Y|X)]. (i) Find the pdf of Y - X.
Let (X,Y) have joint pdf given by f(rw)-y <x, 0 < x < 1, | 0, 0.W., (a) Find the constant c. (b) Find fx (x) and fy(y) (c) For 0 < x < 1, find fy|x=r(y) and My X=r and oỉ x=x (d) Find Cov(X,Y). (e) Are X and Y independent? Explain why.
Let (X,Y) have joint pdf given by I c, \y < x, 0 < x < 1, f(x, y) = { | 0, 0.W., (a) Find the constant c. (b) Find fx(r) and fy(y) (c) For 0 < x < 1, find fy\X=z(y) and HY|X=r and oſ X=z- (d) Find Cov(X, Y). (e) Are X and Y independent? Explain why.
Let (X, Y) have joint pdf given by f(r, y)= < a, 0 < < 0, О.w., (a) Find the constant c (b) Find fx(x) and fy(y) (c) For 0 x< 1, find fyx=r (y) and py|x=x and oyx= (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why
7.5.6 Random variables X and Y have joint PDF fx,y(x, y) = _J1/2 -1 < x <y <1, 1/2 10 otherwise. (a) What is fy(y)? (b) What is fx|v(x\y)? (c) What is E[X|Y = y)?
3. (50 pts) Let (X,Y) have joint pdf given by -{ c, lyl< x, 0 < x < 1, f(x,y) = 0, 0.w., (a) Find the constant c. (b) Find fx(x) and fy(y) (c) For 0< x<1, find fy x-() and pyix- and ox (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
3. (50 pts) Let (X, Y) have joint pdf given by c, y x, 0 < x < 1, f(x, y) 0, o.w., (a) Find the constant c. (b) Find fx(x) and fy (y) (c) For 0 < 1, find fyx=x(y) and pyjx=x and oy Y|X=x (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why.
0 Sy s 1. Let X and Y have joint pdf: fx,y(x, y) = kx(1 – x)y for 0 < x < 1, (a) Find k. (b) Find the joint cdf of (X,Y). (c) Find the marginal pdf of X and of Y. (d) Find Pſy < 81/2],P[X<Y]. (e) Are X and Y independent? (f) Find the correlation and covariance of X and Y. (g) Determine whether X and Y are uncorrelated. (h) Find fy(y|x) (i) Find E[Y|X = x]...