. Suppose that Y is a normal random variable with mean
µ = 3 and variance σ
2 = 1; i.e.,
Y
dist = N(3, 1). Also suppose that X is a binomial random variable
with n = 2 and p = 1/4; i.e.,
X
dist = Bin(2, 1/4). Suppose X and Y are independent random
variables. Find the expected
value of Y
X. Hint: Consider conditioning on the events {X = j} for j = 0, 1,
2.
. Suppose that Y is a normal random variable with mean µ = 3 and variance...
Given random variables X1, X2, Y with E[Y | X1, X2] = 5X1 + X1X2 and E[Y 2 | X1, X2] = 25X2 1X2 2 + 15, find E[(X1Y + X2) 2 | X1, X2]. ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...
Suppose that X is a standard normal random variable with mean 0 and variance 1 and that we know how to generate X. Explain how you would generate Y from a normal density with mean μ and variance σ"? That is, given that we already generated a random variate X from N(0,1), how would you convert X into Y so that Y follows N (μ, σ 2)?
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0...
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0...
Q1 (Joint probability). We know X|Y is a normal random variable with mean Y and variance 2. The probability distribution of Y is a binomial distribution with success probability 0.3 and the number of trials 5. What is the expected value of X?
Let X1 be a normal random variable with mean 2 and variance 3, and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. What is the distribution of the linear combination Y = 2X1 + 3X2?
Suppose that Xi are IID normal random variables with mean 2 and variance 1, for i = 1, 2, ..., n. (a) Calculate P(X1 < 2.6), i.e., the probability that the first value collected is less than 2.6. (b) Suppose we collect a sample of size 2, X1 and X2. What is the probability that their sample mean is greater than 3? (c) Again, suppose we collect two samples (n=2), X1 and X2. What is the probability that their sum...
Assume X is a normal random variable with mean 20 and variance 16, and Y is a Gamma random variable with parameters 5 and 2. In addition X and Y are independent. Construct a box with length L = [X], width W = 2|X|, and height H = Y. Let V be the volume of the box. Calculate the expected value E[V]?
Suppose that X is uniformly selected from the numbers 1, 2, 3 (that is, P(X = i) = 1/3 for i = 1, 2, or 3). Once X is selected, Y is chosen uniformly from the numbers 0, 1, ..., X. Find E[X | Y ]. (c) Compute EX Y 7. Suppose that X is uniformly selected from the numbers 1, 2, 3 (that is, P(X- i) 1/3 for ,X. Find i = 1,2, or 3). Once X is selected,...
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?