Thread: what are the cumulative distribution and pdf "techniques"

1. what are the cumulative distribution and pdf "techniques"

I can't find anything on these techniques in my book. There is plenty of info on the cumulative distribution and the probability density function, but nothing on these "techniques" and no info on how to solve related problems. I am soooo lost. Please help!

1) If X has an exponential distribution with the parameter θ, use cumulative distribution technique to find the probability density of the random variable Y = lnX

2) If the probability density of X is given by f(x) = x/2 on its support 0< x< 2. Find the probability density of Y = X3 by probability density function technique

2. How do you write your exponential density, it does make a difference.

$F_Y(y)=P(Y\le y)=P(\ln X\le y)=P(X\le e^y)=F_X(e^y)$

and if you don't know the cdf of X...

$=\int_0^{e^y}f_X(t)dt$

where I need to know the exact density.
You can write the paramter either way in the exponential.

3. Originally Posted by kesk717
I can't find anything on these techniques in my book. There is plenty of info on the cumulative distribution and the probability density function, but nothing on these "techniques" and no info on how to solve related problems. I am soooo lost. Please help!

[snip]

2) If the probability density of X is given by f(x) = x/2 on its support 0< x< 2. Find the probability density of Y = X3 by probability density function technique
$G(y) = \Pr(Y < y) = \Pr(X^3 < y) = \Pr(X < y^{1/3}) = \int_0^{y^{1/3}} \frac{x}{2} \, dx = ....$ and $g(y) = \frac{dG}{dy}$. There are other details you will also need to fill in.