Question

Let X and Y be continuous random variables with joint distribution function F(x, y), and let...

Let X and Y be continuous random variables with joint distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be functions of X and Y . Prove the following:

(a) E[cg(X, Y )] = cE[g(X, Y )].

(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].

(c) V ar(a + X) = V ar(X).

(d) V ar(aX) = a 2V ar(X).

(e) V ar(aX + bY ) = a 2V ar(X) + b 2V ar(Y ) + 2abCov(X, Y ).

(f) If X and Y are independent, then E[XY ] = E[X]E[Y ].

(g) If X = Y , then Cov(X, Y ) = V ar(Y ).

(h) Cov(X, Y ) = E[XY ] − E[X]E[Y ].

(i) Independence of X, Y =⇒ Cov(X, Y ) = 0.

(j) Cov(X, Y ) = 0 =6⇒ Independence of X, Y .

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose that the joint probability density function of the random variables X and Y is f(x,...
Suppose that the joint probability density function of the random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0 otherwise. (a) Sketch the region of non-zero probability density and show that c = 3/ 2 . (b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1). (c) Compute the marginal density function of X and Y...
19. Let X and Y be continuous random variables with joint pdf: f(x, y) = x−y...
19. Let X and Y be continuous random variables with joint pdf: f(x, y) = x−y for 0 ≤ y ≤ 1 and 1 ≤ x ≤ 2. If U = XY and V = X/Y , calculate the joint pdf of U and V , fUV (u, v).
Suppose X and Y are continuous random variables with joint density function f(x,y) = x +...
Suppose X and Y are continuous random variables with joint density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. (a). Compute the joint CDF F(x,y). (b). Compute the marginal density for X and Y . (c). Compute Cov(X,Y ). Are X and Y independent?
9. Suppose X and Y are continuous random variables with joint density function f(x,y) = x...
9. Suppose X and Y are continuous random variables with joint density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. (a). Compute the joint CDF F(x,y). (b). Compute the marginal density for X and Y . (c). Compute Cov(X,Y ). Are X and Y independent?
1. Let (X; Y ) be a continuous random vector with joint probability density function fX;Y...
1. Let (X; Y ) be a continuous random vector with joint probability density function fX;Y (x, y) = k(x + y^2) if 0 < x < 1 and 0 < y < 1 0 otherwise. Find the following: I: The expectation of XY , E(XY ). J: The covariance of X and Y , Cov(X; Y ).
Let X and Y be continuous random variables with joint density function f(x,y) and marginal density...
Let X and Y be continuous random variables with joint density function f(x,y) and marginal density functions fX(x) and fY(y) respectively. Further, the support for both of these marginal density functions is the interval (0,1). Which of the following statements is always true? (Note there may be more than one)    E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)    E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx    E[Y^3]=∫0 TO 1 y^3 fX(x) dx   E[XY]=(∫0 TO 1 x fX(x)...
Let X and Y be two continuous random variables with joint probability density function f(x,y) =...
Let X and Y be two continuous random variables with joint probability density function f(x,y) = xe^−x(y+1), 0 , 0< x < ∞,0 < y < ∞ otherwise (a) Are X and Y independent or not? Why? (b) Find the conditional density function of Y given X = 1.(
Let X, Y be two random variables with a joint pmf f(x,y)=(x+y)/12 x=1,2 and y=1,2 zero...
Let X, Y be two random variables with a joint pmf f(x,y)=(x+y)/12 x=1,2 and y=1,2 zero elsewhere a)Are X and Y discrete or continuous random variables? b)Construct and joint probability distribution table by writing these probabilities in a rectangular array, recording each marginal pmf in the "margins" c)Determine if X and Y are Independent variables d)Find P(X>Y) e)Compute E(X), E(Y), E(X^2) and E(XY) f)Compute var(X) g) Compute cov(X,Y)
Let X and Y be jointly continuous random variables with joint density function f(x, y) =...
Let X and Y be jointly continuous random variables with joint density function f(x, y) = c(y^2 − x^2 )e^(−2y) , −y ≤ x ≤ y, 0 < y < ∞. (a) Find c so that f is a density function. (b) Find the marginal densities of X and Y . (c) Find the expected value of X
Let X and Y be two continuous random variables with joint probability density function f(x,y) =...
Let X and Y be two continuous random variables with joint probability density function f(x,y) = 6x 0<y<1, 0<x<y, 0 otherwise. a) Find the marginal density of Y . b) Are X and Y independent? c) Find the conditional density of X given Y = 1 /2