# Thread: Find probability density function

1. ## Find probability density function

Suppose that X and Y are independent and each is uniformly distributed on (0, 1). Let U = X +
Y and V = X - Y.
 Sketch the range (the region of non zero probability) of (X, Y) and the range of (U, V).
 Find the density function of (U, V).
 Find the density function of U.
 Find the density function of
I'm not really sure where to start with this question. By being uniform, I assume it means it's mean is 0 and SD is 1? And a normal distribution?
But I couldn't find any information in terms of adding or subtracting them, partly I guess because I have a lot of math to revise for this to make more sense.

If there are any suggestions, or if anyone could prod me in the right direction, i'd appreciate it! =)

2. ## Re: Find probability density function

Hey AshleyCS.

A uniform distribution without any extra information is basically a PDF of P(X = x) = 1 for 0 < x < 1 and 0 everywhere else.

You can find the individual density functions using convolution theorems in probability. With regards to the joint distributions, you should consider all pairs of values for x and y for functions u and v.

A few identities that will be useful will include P(U = u, V = v) = P(U=u|V=v)*P(V=v) = P(V=v|U=u)*P(U=u). You can get the conditionals by comparing the different distributions you obtained above.

How much probability have you covered? What topics are you currently studying?

3. ## Re: Find probability density function

Originally Posted by chiro
Hey AshleyCS.

A uniform distribution without any extra information is basically a PDF of P(X = x) = 1 for 0 < x < 1 and 0 everywhere else.

You can find the individual density functions using convolution theorems in probability. With regards to the joint distributions, you should consider all pairs of values for x and y for functions u and v.

A few identities that will be useful will include P(U = u, V = v) = P(U=u|V=v)*P(V=v) = P(V=v|U=u)*P(U=u). You can get the conditionals by comparing the different distributions you obtained above.

How much probability have you covered? What topics are you currently studying?