# Thread: probability distribution and density function

1. ## probability distribution and density function

Hi,

How do you determine the probability distribution function F of the stochastic variable U:= X+Y where X and Y are independent uniformly distributed over [0,1]?

My thought was to solve this by first determining the density function f of U
So f wil be the convolution product of the density functions of X and Y
So f(u)=(g * h)(u) where g and h are the density functions of X and Y

Then Prob(U<= c) = F(c) = integral (from 0 to c) f(x)dx

Is this the correct way of determining F?

And if I want to find the density function of Z:=X-Y is that also possible by using the convolution of the density function of X and -Y? Is -Y uniformly distributed?

I hope anyone can help me out!

Thanks!

Kind regards,
Joolz

2. Originally Posted by Joolz
Hi,

How do you determine the probability distribution function F of the stochastic variable U:= X+Y where X and Y are independent uniformly distributed over [0,1]?

My thought was to solve this by first determining the density function f of U
So f wil be the convolution product of the density functions of X and Y
So f(u)=(g * h)(u) where g and h are the density functions of X and Y

Then Prob(U<= c) = F(c) = integral (from 0 to c) f(x)dx

Is this the correct way of determining F?

And if I want to find the density function of Z:=X-Y is that also possible by using the convolution of the density function of X and -Y? Is -Y uniformly distributed?

I hope anyone can help me out!

Thanks!

Kind regards,
Joolz
Yes convolution is what you use to find the density of the sum of two RVs,

If Y ~U(0,1), then -Y ~U(-1,0)

CB

3. Originally Posted by Joolz
Hi,

How do you determine the probability distribution function F of the stochastic variable U:= X+Y where X and Y are independent uniformly distributed over [0,1]?

My thought was to solve this by first determining the density function f of U
So f wil be the convolution product of the density functions of X and Y
So f(u)=(g * h)(u) where g and h are the density functions of X and Y

Then Prob(U<= c) = F(c) = integral (from 0 to c) f(x)dx

Is this the correct way of determining F?

And if I want to find the density function of Z:=X-Y is that also possible by using the convolution of the density function of X and -Y? Is -Y uniformly distributed?

I hope anyone can help me out!

Thanks!

Kind regards,
Joolz