Let X be a random variable with E[X] = 2, Var(X) = 4. Compute the expectation and variable of 3 - 2X.
Let X be a random variable with E[X] = 2, Var(X) = 4. Compute the expectation...
O RANDOM VARIABLES AND DISTRIBUTIONS Expectation and variance of a random variable Let X be a random variable with the following probability distribution: Value x of X P(X-) 0.35 0.40 0.10 0.15 10 0 10 20 Find the expectation E (X) and variance Var(X) of X. (If necessary, consult a list of formulas.) Var(x) -
6. Let X be an exponential random variable with parameter 1 = 2. Compute E[ex]. = 7. Consider a random variable X with E[X] u and Var(X) 02. Let Y = X-4. Find E[Y] and Var(Y). The answer should not depend on whether X is a discrete or continuous random variable.
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...
What is Var[3X]? Let X be a random variable such that Var[X] = 5 and E[X] = 4.
Problem 1. (a) Let X be a Binomial random variable such that E(X) 4 and Var(x) 2. Find the parameters of X (b) Let X be a standard normal random variable. Write down one function f(t) so that the random variable Y-f(X) is normal with mean a and variance b.
3. Let X be a continuous random variable with E(X)-μ and Var(X)-σ2 < oo. Suppose we try to estimate μ using these two estimators from a random sample X, , X,: For what a and b are both estimators unbiased and the relative efficiency of μι to is 45n?
S 4, with the density function 1. (10 points) Let X be a continuous random variable on 3 S f(x) = 2x - 3). a/ CNculate Pr(3.2 S X) and Pr(3 < X) b/ Find E(X) and Var(x) 2. (10 points) For any number A, verify that fæ) -e-, 12 A is a density function. Compute the assocuated cumulative distribution for X
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] = ax +b and Var[Y|X =x] = 1 a) compute E[Y] b) compute Var[Y] c) Find E[XY]
Let x be a random variable with the following probability distribution: Value x of X -2 - 1 0 0 0 1 0 P(X-X) 0.10 .30 .20 .40 Find the expectation E (x ) and variance Var (x) of X. (If necessary, consult a list of formulas.) ( x 5 ? Var (x) - 0
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....