# Distribution of weighted Normal Distributions

1. Sep 15, 2011

### TOOP

You created a random number generator that works as follows:
With probability p it selects a number X from the standard normal distribution N(0,1), and
with complimentary probability (1-p) it selects a random number X from an off-central
normal distribution N(5, 1). Write the distribution function of X.

How would you attempt this.
Obviously the variance increases.
and the mean is a weighted average of the two.
but as far as getting fx(x) I am stumped.
Is it correct to add the two distributions together and simplify?
(p)*N(0,1) + (1-p)*N(5,1)
using the gaussian equation?

2. Sep 15, 2011

### lanedance

looks reasonable to me - when you have a discrete probability to realise a given distribution, you can just add the distributions together weighted by their probabilties

if it doubt try it, this would be pretty simple to implement in excel as a check

3. Sep 15, 2011

### Ray Vickson

Yes, it is correct. However, it won't simplify. The final result is NOT itself a Gaussian, or anything like it.

RGV