Find the CDF and PDF for Y = |X| if X is uniformly distributed in the interval (

Question

Find the CDF and PDF for Y = |X| if X is uniformly distributed in the interval (-1, 1)

Explanation / Answer

First consider x > 0. The area to the left of x isone-half (the area of the triangle whose vertices are (-1,-1),(-1,0), and (0,0)) + the area of the triangle whose vertices are(0,0), (x,0), and (x,x), namely x-squared divided by 2. Check: when x=1, the cumulative probability is 1, as it mustbe. The density function is -x for x from -1 to zero, and xfor x from zero to +1. For x < 0, the area to the left ofx is one-half (that being the area of the mirror-image trianglewhose vertices are (-1,-1), (0,0), and (-1,0)) minus the area of atriangle whose vertices are (x,x), (0,0), and (x,0).

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