Let X, Y, and Z have the joint probability density function F(x,y,z) = { kx2 yz, 0 < x, y < 1, 0 < z < 2, a) Find k b. Find P(X < ¼, Y > ½, 1 < Z < 2)
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- Let (X, Y ) be a random vector with joint density function of the form:f(X,Y )(x, y) = 24xy, 0 < x < 1, 0 < y < 1, x + y < 10 , otherwise(a)Compute the probability P((X, Y ) ∈ [0, 1/2] × [0, 1/2])Let (X, Y ) be a random vector with joint density function of the form:f(X,Y )(x, y) = 24xy, 0 < x < 1, 0 < y < 1, x + y < 10 , otherwise(c)Compute the conditional probability P(X < 1/6|Y=1/2) and the conditional expectation E(X | Y = y)Let X and Y be two independent random variables, X ∼ Γ(α, λ) and Y ∼ Γ(β, λ). Find the joint probability density function f(Z,W)of the vector (Z, W)(b) Show that Z and W are independent(c) Show that Z ∼ Γ(α + β, λ) and W ∼ B(α, β)
- If the joint probability density of X and Y is given by f(x, y)=2 for x>0, y>0, x+y<1, and f(x, y) = 0 elsewhere, find(a)P(X<=1/2, Y<=1/2);(b)P(X+Y>2/3);(c)P(X>2Y)Suppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).
- Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentLet (X, Y ) be a random vector with joint density function of the form:f(X,Y )(x, y) = 24xy, 0 < x < 1, 0 < y < 1, x + y < 10 , otherwise(d) Check that E(E(X | Y )) = E(X).Let the joint density function of (X,Y) be: f(x,y) = (x+y) / 4, 0 < x < y < 2. a) Find fX|Y(x|y). b) Compute the conditional probabilities P(X < 1/2 | Y = 1) and P(X < 3/2 | Y = 1).
- Let D = {(x, y) ∈ R^2: x^2 + y^2 ≤ 1} be the unit disk and (X, Y ) be uniformly distributed on D. Find the probability density function of W for 1. W = X + Y .Let X be a continuous random variable with density function f(x) = {2x if x ∈ [0,1] {0 otherwise Compute E[X] and E(X2).Suppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?