Consider a random variable X with the following properties E[X] = 20 and var(X) = 2....
Consider a random variable X with the following properties E[X] - 10 and var(X) - 9. Consider a new random variable such that Y-1-5X Calculate the following (a) EY] - (b) var(Y) = 5
Recall that the variance of a random variable is defined as Var[X]=E[(X−μ)2], where μ = E[X]. Use the properties of expectation to show that we can rewrite the variance of a random variable X as Var [X]=E[X^2]−(E[X])^2 Problem 3. (1 point) Recall that the variance of a random variable is defined as Var X-E(X-μ)21, where μ= E[X]. Use the properties of expectation to show that we can rewrite the variance of a random variable X as u hare i- ElX)L...
5. Suppose X and Y are random variables such that E(X)=E(Y) = θ, Var(X) = σ and Var(Y)-吆 . Consider a new random variable W = aX + (1-a)Y (a) Show that W is unbiased for θ. (b) If X and Y are independent, how should the constant a be chosen in order to minimize the variance of W?
2. Explain in words, and words only, the following properties of expected values. NOTE: X and Y are random variables and k is a constant. (a) E(k) = k (b) E(X+Y) = E(X) + E(Y) (c) E(X/Y) + E(X)/E(Y) (d) E(X+Y) E(X)*E(Y) (unless what?) (e) E(X2) # (E(X)]? (1) E(kX) = E(X) 3. For random variable X with mean H. variance is defined var(X) = Ef(X-M.)'. Show how variance can be expressed only in terms of E(X) and E(X). 4....
(1 point) For a random variable X, suppose that E[X] = 2 and Var(X) = 3. Then (a) E[(5 + x)2) = (b) Var(2 + 6X) =
5. Suppose X is a normally distributed random variable with mean μ and variance 2. Consider a new random variable, W=2X + 3. i. What is E(W)? ii. What is Var(W)? 6. Suppose the random variables X and Y are jointly distributed. Define a new random variable, W=2X+3Y. i. What is Var(W)? ii. What is Var(W) if X and Y are independent?
2. (7 pt) Recall that the variance of a random variable X is defined by Var(X) - E(X - EX)2. Select all statements that are correct for general random variables X,Y. Throughout, a, b are constants. ( Var(X) E(X2) (EX)2 ( ) Var(aX + b) = a2 Var(X) + b2 Var(aXb)a Var(X)+b ( ) Var(X + Y) = Var(X) + Var(Y) ) Var(x) 2 o ) Var(a)0 ( ) var(x") (Var(X))"
2. Explain in words, and words only, the following properties of expected values. NOTE: X and Y are random variables and k is a constant. (a) E(k) = k (b) E(X+Y) = E(X) + E(Y) (c) E(X/Y) + E(X)/E(Y) (d) E(X+Y) E(X)*E(Y) (unless what?) (e) E(X2) # (E(X)]? (1) E(kX) = E(X) 3. For random variable X with mean H. variance is defined var(X) = Ef(X-M.)'. Show how variance can be expressed only in terms of E(X) and E(X). 4....
Let X, Y, Z be random variables with these properties: · E[X] = 3 and E[X²] = 10 Var(Y) = 5 E[Z] = 2 and E[Z2] = 7 • X and Y are independent E[X2] = 5 Cov(Y,Z) = 2 Find Var(3X+Y – Z).
Let Xand Y be random variables with the following properties: Settings Var(X)-0%-0.3 Var(Y)-Cy-0.5 Find the following: E(SX-3Y) if needed round your solution to 2 decimal places