# How to put two normal distributions together

• Jan 17th 2009, 01:57 PM
Cactor
How to put two normal distributions together
If I have a box with 2 types of light bulbs, A and B. The lifespan of each type of lightbulb follows a gaussian distribution A ~ N(1000h, 5h), and B ~ N(990h, 4h). I know that 70% of them are A and the other 30% are B. How can I know the normal distribution of my box? I think it's obvious that the new mean will be μA*0,7+μB*0,3, but how can I know the new standard deviation? Or maybe I just can't?
• Jan 17th 2009, 02:33 PM
mr fantastic
Quote:

Originally Posted by Cactor
If I have a box with 2 types of light bulbs, A and B. The lifespan of each type of lightbulb follows a gaussian distribution A ~ N(1000h, 5h), and B ~ N(990h, 4h). I know that 70% of them are A and the other 30% are B. How can I know the normal distribution of my box? I think it's obvious that the new mean will be μA*0,7+μB*0,3, but how can I know the new standard deviation? Or maybe I just can't?

You have \$\displaystyle Y = 0.7 X_A + 0.3 X_B\$ and I assume that \$\displaystyle X_A\$ and \$\displaystyle X_B\$ are independent. Then \$\displaystyle Var(Y) = (0.7)^2 Var(X_A) + (0.3)^2 Var(X_B)\$.

Aside: There are several well known approaches to proving that a linear combination of a pair of independent random variables is itself normal.