# Resultant of two distribution (Normal)

#### sandiphw

##### New Member
If there are two normal distribution having mean and s.d M1,SD1 and M2,SD2 respectively, what would be the mean and s.d of the resultant these two distribution. I usderstand that the resultant of these two will also be a normal distribution.

For example, the particle size distribution in two containers are (M1, SD1) and (M2,SD2) and both are mixing to a third container, what would be the mean, standard deviation of the mixer in third container.

Similarly, if two systems are in series, how can I find the resultant of these two distribution. For ex, there are two crushing mill, both have a crushing size distribution. From first crusher material goes to second one. How can I find the distribution of final output.

Thanks

E

#### elnaz

##### Guest
Hello
your answer is dependent to type of function that it is between two normal variables
is your function y=x1+x2 that x1 and x2 are normal variables and y is Resultant of two normal variables or another things
but the first you say about type of two normal function Resultant then
we could speak about its sd and mean

#### sandiphw

##### New Member
Dear Elnaz,

Thanks for your reply and sorry if I couldn't explain my problem properly. I would like to request you to have a look at the attachment (diagram.pdf). It is a problem I am facing in real life. Say, there are two containers having some samples of quality following a normal distribution having mean & std dev M1, SD1 & M2, SD2 respectively. This two samples are mixing in a third container. The quality of the mixed samples must follow a normal distribution. How to find out the mean & std dev of the quality of the mixed sample?

In second case, the two system are in series, what would be the mean & std dev of the quality of the mixed sample?

Hope I can explain the my requirement.

#### JohnM

##### TS Contributor
Unfortunately this is not a straightforward calculation....

In the upper part of your diagram, there is no guarantee that the mixed distribution will be normally distributed. An extreme example to make my point: what if you mixed a container of sand (small particles) with a container of marbles (large particles)? The resultant distribution will be bi-modal (two "humps" separated by a large amount). The mean will simply be the weighted mean of the two original means. I don't know if there is a general formula to determine the mixed standard deviation.

In the second situation, I believe the resultant mean will be the product of the first two means, but I'm not 100% positive about that.

#### sandiphw

##### New Member
Thanks JohnM for you views. But in real life, people must be used some logics to get the result. Because I have seen peoples to use it in a vriety of form in Market Research field. If you can fine some sorts of solutions please let me know.

Sandip