# probability function of sum of multiple normal distributions

#### Sf26

##### New Member
I have four normally distributed random variables which may/may not be correlated. I want to find the probability distribution function of the sum? For example, if means were 10,15,20,10 and sd=4,2,3,5 respectively - what would the probability the sum of the variables would be greater than lets say 70?

I am trying to do this in R. Would this be just the sum of the means and square root of the sum of the variance as the SD and draw from that distribution? Or is there something I am missing....any help would be appreciated!

Thanks!

#### BGM

##### TS Contributor
The sum will follow a normal distribution if they are jointly follow a multivariate normal.

Your calculation is correct if they are uncorrelated.

#### Dason

That is a big "if" though. If they are correlated it will still follow a normal distribution with that same mean but the standard deviation will be different.

#### hlsmith

##### Omega Contributor
I have four normally distributed random variables which may/may not be correlated.
Or do you have parameters for these four variables? I am guessing you don't have the actual data, right?

#### Sf26

##### New Member
That is a big "if" though. If they are correlated it will still follow a normal distribution with that same mean but the standard deviation will be different.

I do have the actual data with sample means and deviations. I was concerned that if you had two normal distributions - one has a peak of lets say 50 another has a peak of 5 , are you able to draw from that distribution as normal even though the combination are two separate "humps"?