Find the value of the constant c. Find P(YX). Find P(Y = X).

Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2  with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?

Suppose X and Y are independent random variables with E(X) =2, E(Y)=3,V(X)=4,V(Y)=16. Finda)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y) d)COV(X,5X-Y)

Suppose that the random variables X and Y are independent with Var(X)=8 and Var(Y)=6. Calculate Var(5X−7Y+17)

Suppose X is a random variable such that E(X) = 90 and Var(X) = 7. Compute thefollowing: E(4X + 12) Var(−X) SD(4X)E(X^2) E(X + 3)^2

For the random variables X,Y Cov(X,Y) = -0.9 if Z=3-X then what is Cov(Z,Y)=???

2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive constants such that a+b= 1, then (i) Find mgf. (ii) If E(X) = m₁, E(X2) = m2, show that m₂ = m₁(2m₁ + 1).

2. Suppose that X is a random variable with E(X) = 6 and Var(X) = 4. Find the following: a) Е(4X + 1) b) Var(3X-2) c) E(X²).

3. Consider two random variables X₁ and X2 whose joint pdf is given by for x₁0, x2 > 0, x₁ + x2 < 2, { Find the pdf of Y = X₂ - X₁. f(x1, x₂) = NIT otherwise.

3. Let the random variable X have the pmf f(x) = a) E(X) b) E(X²) c) E(3X²2X + 4) (x+1)² 9 for x = -1,0,1. Compute