Help with probability distribution function question

[TPDC130]

Homework Statement

Let X and Y be two independent random variables with the same probability density funtion over:

f(x) = {1/a if x € [0,a]
{0 if x=0

Find the density distribution of a) X + Y and b) X*Y

Homework Equations

The Attempt at a Solution

Ok, my initial thoughts are for:

a) That f(X+Y) is simply the product of each of their densities, which would result in 1/(a^2)

b) f(X*Y) is the sum of their individual distributions. i.e. 2/a


This seems a little too simple for me and I think I am looking at it from the wrong perspective.

Any thoughts or advice?

 

[dongo]

No, i don't think this is right. Just consider that the two random variables are idependently and identically distributed w.r.t the distribution you mentioned above. So $X + Y$takes its values in the interval $[0, 2a]$. What you were referring to is the Gaussian distribution where the result concerning a would be valid. In this case however, since the points are distributed according to (1) a continuous distribution which is (2) uniform, we conclude that if the two random variable $X, Y$ also are distributed according to this distribution, then there sum has to be distributed (again uniformly) with probability density $1/2a$. Converesely, take question b. You may regard X and Y as some sort of random coordinates which give you point lying a sqaure with side length a. Again, recalling that we deal with uniform distributions, we find the probability density function to be constant $1/a^{2}$. The problem is that one has to distinguish between uniform distributions and the standard distributions, i.e., Gaussian, binomial, geometric, multinomial, Poisson, exponential.

 

[Ray Vickson]

You can get the density f(z) of X+Y by differentiating the CDF F(z) = Pr{X+Y <= z}, and the latter is an integral over the two-dimensional region {x+y <= z, 0 <= x,y <= a}. Or, Google "convolution".

RGV