Prof. David Draper
Department of Statistics
University of California, Santa Cruz
STAT 131: Quiz 6 50 total points
Name:
In a problem you’re working on, you need to simulate random draws from the following
PDF for a continuous random variable Y :
fY (y) =
{
3 (1− y)2 if 0 ≤ y ≤ 1
0 else
}
= 3 (1− y)2 I(0 ≤ y ≤ 1) . (1)
(a) Sketch the PDF in equation (1) for y in the interesting range [0, 1]. 10 points
(b) Work out the CDF FY (y) for Y , specifying its values for all −∞ < y < +∞, and
sketch it in the interesting range 0 ≤ y ≤ 1. NB Please make the calculation by
hand, to avoid the nonsense mentioned in (c) below. 10 points
(c) Work out the inverse CDF (quantile function) F−1Y (p) for Y , specifying its values for
all 0 < p < 1, and sketch it for p in that range. Warning: Wolfram Alpha is capable
of producing a nonsensical answer to this question; you can avoid that by doing the
computation in (b) by hand and using the result to get your answer for (c), also by
hand. 10 points
1
(d) Building on your result in part (c), assuming that you have access to a Uniform(0, 1)
pseudo-random number generator, explicitly specify how you could generate IID
pseudo-random draws from the PDF in equation (1). 10 points
(e) Once you have your pseudo-random sample in part (d), briefly explain how you could
graphically check whether it really is a random sample from the PDF in equation
(1). 10 points