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(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let ...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) =...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) = | Fy(x) · fx (x) dx -oo where Fy is the CDF of Y and fx is the PDF of X [hint: P[Y E A] = S.P(Y E A|X = x) · fx(x) dx]. Rewrite the above equation as an expectation of a function of X, i.e. P(Y < X) = Ex[•]. Use the above relation to compute P[Y < X] if X~Exp (2)...
The answer will be: Consider two independent random variables X and Y. Let fx(x) 1-2 if...
The answer will be:
Consider two independent random variables X and Y. Let fx(x) 1-2 if 0 〈 x 〈 2 2-2y for 0-y 〈 l and 0 otherwise. Find the probability density function of X + Y. and 0 otherwise. Let Jy(y) 42.4 If 0-a-l then Íx+y(a) = 2a--a2+ ). If 1- a < 2 then 3 213 then jx+Y(a then fx+y(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...
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...
Let X and Y be continuous random variables with joint pdf fx y (x, y)-3x, 0...
Let X and Y be continuous random variables with joint pdf fx y (x, y)-3x, 0 Sy and zero otherwise. 2. sx, a. What is the marginal pdf of X? b. What is the marginal pdf of Y? c. What is the expectation of X alone? d. What is the covariance of X and Y? e. What is the correlation of X and Y?
The random variables X and Y have joint PDF fX,Y(x,y) = {12x2y 0<=x<=c; 0 <= y...
The random variables X and Y have joint PDF fX,Y(x,y) = {12x2y 0<=x<=c; 0 <= y <= 3 { 0 otherwise (a) FInd the value of C (b) Find the PDF fW(w) where W = X / Y (c) Find the PDF fZ(z) where Z = min(X,Y)
The random variables X and Y have the joint PDF fx,y(x,y)=0.5, if x>0 and y>0 and...
The random variables X and Y have the joint PDF fx,y(x,y)=0.5, if x>0 and y>0 and xtys2, and 0 otherwise. Let A be the event Ys1) and let B be the event (Y>X). (You can use rational numbers like 3/5 for your answers.) 1. Calculate P(BIA). 2. Calculate fxıy(xlO.9) fxIY(0.39820710.9) 3. Calculate the conditional expectation of X, given that Y=1.8 4, Calculate the conditional variance of X, given that Y=1.4 5. Calculate fxlB(x) fXIB(0.11) 6. Calculate E[XY]. 7. Calculate the...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
7.5.6 Random variables X and Y have joint PDF fx,y(x, y) = _J1/2 -1 < x...
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)?
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) = | Fy(x) · fx (x) dx -oo where Fy is the CDF of Y and fx is the PDF of X [hint: P[Y E A] = S.P(Y E A|X = x) · fx(x) dx]. Rewrite the above equation as an expectation of a function of X, i.e. P(Y < X) = Ex[•]. Use the above relation to compute P[Y < X] if X~Exp (2)...
The answer will be:
Consider two independent random variables X and Y. Let fx(x) 1-2 if 0 〈 x 〈 2 2-2y for 0-y 〈 l and 0 otherwise. Find the probability density function of X + Y. and 0 otherwise. Let Jy(y) 42.4 If 0-a-l then Íx+y(a) = 2a--a2+ ). If 1- a < 2 then 3 213 then jx+Y(a then fx+y(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...
Let X and Y be continuous random variables with joint pdf fx y (x, y)-3x, 0 Sy and zero otherwise. 2. sx, a. What is the marginal pdf of X? b. What is the marginal pdf of Y? c. What is the expectation of X alone? d. What is the covariance of X and Y? e. What is the correlation of X and Y?
The random variables X and Y have the joint PDF fx,y(x,y)=0.5, if x>0 and y>0 and xtys2, and 0 otherwise. Let A be the event Ys1) and let B be the event (Y>X). (You can use rational numbers like 3/5 for your answers.) 1. Calculate P(BIA). 2. Calculate fxıy(xlO.9) fxIY(0.39820710.9) 3. Calculate the conditional expectation of X, given that Y=1.8 4, Calculate the conditional variance of X, given that Y=1.4 5. Calculate fxlB(x) fXIB(0.11) 6. Calculate E[XY]. 7. Calculate the...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
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)?
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
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