Question

2. Let X be a uniform random variable over the interval (0, 1). Let Y = X(1-X). a. Derive the pdf for Y . b. Check the pdf you found in (a) is a pdf. c. Use the pdf you found in (a) to find the mean of Y . d. Compute the mean of Y by using the distribution for X. e. Use the pdf of Y to evaluate P(|x-1/2|<1/8). You cannot use the pdf for X. f. Use the pdf of X to evaluate P(|X-1/2|<1/8).

Answer 1

a. We have Y = X(1-X),

So as X ~ Unif(0,1) ; X (0,1) X(1-X) = X- > 0

We first find c.d.f of Y,

=1- P(X is in between the roots of    which has real roots iff y < 1/4

So, =

hence pdf is obtained by differentiating cdf,

for   

b.      which on putting limits gives 1. Hence f is indeed a pdf.

c. Mean using distribution of Y

d. Mean using distribution of X

We are advised to solve only four parts. So it would be advisable to pose the next two parts of the problem as a separate problem.

Thank you! Postive feedback is highly appreciated!