Suppose that X is a standard normal random variable with mean 0 and variance 1 and...
. 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. 8....
Let X be a normal random variable with mean 0 and variance σ^2. Find the density for |X|.
2. Suppose that you can draw independent samples (U,, U2,U. from uniform distribution on [0,1]. (a) Suggest a method to generate a standard normal random variable using (U, U2,Us...) Justify your answer. b) How can you generate a bivariate standard normal random variable? (Note that a bivariate standard normal distribution is a 2-dimensional normal with zero mean and identity covariance matrix.) (c) What can you suggest if you want to generate correlated normal random variables with covariance matrix Σ= of...
Let X variable Y by be a normal random variable with mean 0 and variance 1. We define the random y2 if x 20, Y= (a For t E R, compute Mr()-Elen'], the moment generating function of Y. Compute EY
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
Suppose X is a normal random variable with mean μ = 100 and standard deviation σ = 10. Find a such that P(X ≥ a) = 0.04. (Round your answer to one decimal place.) a =
Suppose X is a normal random variable with mean μ = 70 and standard deviation σ = 5. Find a such that P(X ≥ a) = 0.01. (Round your answer to one decimal place.) a =
1. The random variable X is Gaussian with mean 3 and variance 4; that is X ~ N(3,4). $x() = veze sve [5] (a) Find P(-1 < X < 5), the probability that X is between -1 and 5 (inclusive). Write your answer in terms of the 0 () function. [5] (b) Find P(X2 – 3 < 6). Write your answer in terms of the 0 () function. [5] (c) We know from class that the random variable Y =...
3. Suppose that the random variable X is an observation from a normal distribution with unknown mean μ and variance σ (a) 95% confidence interval for μ. (b) 95% upper confidence limit for μ. (c) 95% lower confidence limit for μ. 1 . Find a
2. Suppose Yi,.. narei normal random variables with normal distribution with unknown mean and variance, μ and or. Let Y-욤 Σ;..x. For this problem, you may not assume that n is large. (a) What is the distribution of Y? (b) what is the distribution of z-(yo), (en, (n-) (c) what is the distribution of (n-p? (d) What is the distribution of Justify your answer. (e) Let Zi-(ga)' + (-)' + (yo)", z2 = (속)' + (n-e)' what is the distribution...