# Thread: distribution of sum of random variables

1. ## distribution of sum of random variables

if random variables X1...Xn have a uniform distribution, would it be correct to assume that the sum of Xi has also the uniform distribution?
and if so how can i prove it?

i think Laplas transformation can be used... am i right?

if random variables X1...Xn have a uniform distribution, would it be correct to assume that the sum of Xi has also the uniform distribution?
No, it isn't. The density becomes more and more complicated when $\displaystyle n$ increases; I think I read something about this in Feller vol.1. Are all your variables uniform on the same interval (like, on $\displaystyle [0,1]$)? Do you need an explicit formula?

3. Originally Posted by Laurent
No, it isn't. The density becomes more and more complicated when $\displaystyle n$ increases; I think I read something about this in Feller vol.1. Are all your variables uniform on the same interval (like, on $\displaystyle [0,1]$)? Do you need an explicit formula?
I'll add to the list: Are all the variables independent? (If so, a moment generating function approach might be the best way forward ....)

4. hi all,

sorry for mess, i'll try to calrify more.

the full question is:

If 10 fair dice are rolled, approximate the probability that the sum of values obtained (which is from 20 to 120) is between 30 and 40 inclusive.

so i figured that

X1....Xn are uniform[1,6] variables, and i need to find the distribution of
Y=X1+X2+....Xn.

1)if i'm right what is the distribution of Y?
2) how do i compute P(30<Y<40)?

hopes this is more clear, thanks...

5. Hello,
hi all,

sorry for mess, i'll try to calrify more.

the full question is:

If 10 fair dice are rolled, approximate the probability that the sum of values obtained (which is from 20 to 120) is between 30 and 40 inclusive.

so i figured that

X1....Xn are uniform[1,6] variables, and i need to find the distribution of
Y=X1+X2+....Xn.

1)if i'm right what is the distribution of Y?
2) how do i compute P(30<Y<40)?

hopes this is more clear, thanks...
Yes, it clarifies a lot !

Because X1,...,Xn are uniform over {1,2,3,4,5,6}, which is not [1,6], the set of all real numbers that are between 1 and 6.

You're working here with discrete random variables, not continuous variables.

6. The sum of two uniform (0,1) is a traingular density on (0,2).
It's x on (0,1) and 2-x on (1,2) and zero elsewhere.
AND the MGF of uniforms are worthless, not the way to approach this.
The classic way is via the CDF of the sum.

The word approximate implies the CLT, but here you have a small sample.
So, I'm not sure what you're supposed to do.
You can obtain the distribution of the sum of ten of these rvs.

hi all,

sorry for mess, i'll try to calrify more.

the full question is:

If 10 fair dice are rolled, approximate the probability that the sum of values obtained (which is from 20 to 120) is between 30 and 40 inclusive.

so i figured that

X1....Xn are uniform[1,6] variables, and i need to find the distribution of
Y=X1+X2+....Xn.

1)if i'm right what is the distribution of Y?
2) how do i compute P(30<Y<40)?

hopes this is more clear, thanks...
Despite the fact that you only have the sum of 10 variables, my guess is that you are supposed to assume that the distribution of the sum is approximately Normal (i.e. Gaussian) and apply the Central Limit Theorem. So you will need to find the mean and variance of the sum before proceeding -- which is pretty easy. But note that you are dealing with a discrete uniform distribution over 1, 2, ..., 6, not a continuous Uniform[1,6] distribution; it makes a difference.

(It is possible to find the exact distribution of the sum, but messy -- hence my guess that you are supposed to assume approximate normality. And the problem statement did say "approximate the probability".)

8. see attached,