Hi there hazeleyes,
Here's a kick off...
for the mean
hey guys, I'm just wondering if you guys can help me with this question, thank you in advance.
A random variable, X, has a probability density function (p.d.f.) given by
Y is a random variable such that Y = X^2.
Show that Y has an exponential distribution and state its mean.
There are several approaches that can be taken.
The OP has carelessly neglected to mention that the support of X is (so that f(x) = 0 for x < 0). The support of Y will be .
The most basic approach to finding the pdf of Y is to calculate the cdf of Y and then recall that the pdf is the derivative.
Differentiating this is simple - either integrate (use a substitution) directly and then differentiate, or use the chain rule and the Fundamental Theorem of Calculus.
You should get for and zero elsewhere. Calculating the mean of Y is trivial.
Some trivia: The random variable X follows a Weibull distribution (http://en.wikipedia.org/wiki/Weibull_distribution) with parameters and . Since it is well known that Y follows an exponential distribution.
I follow mr. fantastic's logic up to the point of G(y) = P(-y1/2 <= X <= y1/2)
But shouldn't this mean that G(y) = the integral from 0 to y1/2 of 2e-2x dx = 1 - e-2y1/2?
and therefore, g(y) = dG/dy = y-1/2 e-2y1/2?
Sorry, regarding my last post I realized I did not read the original post carefully enough - I thought the original pdf was f(x) = 2e-2x. In the process however, I think I learned that where X~exp(lambda), Y= X2 is also exp(lambda). Is that correct?