# Question

Let X be N(0, 1). Find the pdf of Y = |X|, a distribution that is often called the half-normal. Hint: Here y ∈ S y = {y : 0 < y < ∞}. Consider the two transformations x1 = −y, −∞ < x1 < 0, and x2 = y, 0 < y < ∞.

## Answer to relevant Questions

Let X have the Find the pdf of Y = X2. Let W1, W2 be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, (W1 + W2)/2. (a) Show that the pdf of X1 = (1/2)W1 is (b) Let Y1 = X1 + X2 = W and Y2 = X1, where X2 = (1/2)W2. ...Let X1, X2, X3 denote a random sample of size n = 3 from a distribution with the geometric pmf (a) Compute P(X1 = 1, X2 = 3, X3 = 1). (b) Determine P(X1 + X2 + X3 = 5). (c) If Y equals the maximum of X1, X2, X3, find P(Y ≤ ...Let X1 and X2 be a random sample of size n = 2 from a distribution with pdf f(x) = 6x(1 − x), 0 < x < 1. Find the mean and the variance of Y = X1 + X2. Generalize Exercise 5.4-3 by showing that the sum of n independent Poisson random variables with respective means μ1, μ2, . . . , μn is Poisson with mean μ1 + μ2 + · · · + μn.Post your question

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