Probability Density Function and Cumulative Distribution Function

#1
Hi everybody,

If anyone has experience working with probability density functions (pdf), please look these sample problems over and if you could, give me a few pointers as how to approach them.

I am currently enrolled in a course called Advanced Probability and Statistics. I keep up with all the reading and pay attention in class but I still don't fully comprehend this particular material. Therefore, any insight would be greatly appreciated. I am hoping to learn from a different perspective. My sample problems are below.

Problem 1: The random variable X has a pdf given by fx(x) = 4x^3 if 0< x <1 and zero otherwise. Find the CDF (cumulative distribution function) and use it to determine the pdf of the following random variable Z = ln(X).

Problem 2: Let X have pdf fx(x) = x^2/24 if -2< x < 4 and zero otherwise. Find the pdf of Y = X^2.

Again, thank you for your time and interest.

Kookie
 

Dason

Ambassador to the humans
#2
You've explained that you're having trouble here but it would help us out a lot if you explained exactly what is giving you trouble. What about finding a CDF is tripping you up? What parts don't you understand.
 
#3
Hi everyone,

For the members that have knowledge and experience in working with pdf and CDF, Perhaps this will facilitate the process.

For problem one, I know the first part (how to find the CDF of X). You simply integrate the given pdf to obtain the CDF of X. What I'm uncertain about is how to use the 'found' CDF of X to find the pdf of the random variable Z. I know you're supposed to replace something with another thing and that's where I'm unclear.

As for problem two, is it pretty much the same procedures as one?

I hope this helps a bit. I'm not just looking for help without putting my own effort in. I have an idea of the material, just not as clear and thorough as I would like. Therefore, any insight would be helpful. Thank you, all.
 
#4
Thank you for your interest in helping me, Dason. At the same time you posted, I also posted a different message explaining what I'm having difficulty with. Please see if you can give me any pointers. Thanks, once again,
 
#5
Hi all,

Can somebody please tell me how to find the pdf of a second random variable after I have found the CDF of the first random variable from its pdf? Any insight at all would be greatly appreciated.
 
#6
firstly this is not a one to one transformation
you have to find the jacobian of transformation you have two values of jacobian one is +ve value(say j1) and the 2nd one is -ve value(say j2)
to make the pdf of new variable Y you should break your limits -2<x<4
to -2<x<0 and 0<x<4 and then integrate your f(x) with limit 0 to 4 and multiply it with j2
and similarly integrate f(x) with range 0 to 16 and multiply it with j2
and then add these two answer this is your new pdf of y with range 0<y<16