Question

Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find:

a) The PDF fY (y)

b) The mean E[Y ]

c) Here X is uniform in (0, 1), but now you are asked to find a functiong(·) such that the PDF of Y = g(X) is

?2y 0≤y<1fY (y) = 0 otherwise

Answer 1

a)

We are given that X ~ N(0,1) and Y = |X|.

We need to find the pdf of Y.

First we find the cdf of Y:

Now, to find the pdf of Y, we just differentiate the cdf w.r.t. y. Thus, we get:

b)

c)

For , the pdf of Y is given by:

Thus, for , the cdf of Y is given by:

Thus,

which is the required function.

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