Help with probability distribution function question

1. Jul 28, 2011

TPDC130

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 28, 2011

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 continous 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.

3. Jul 28, 2011

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