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Math Help - How to put two normal distributions together

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    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?
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    Quote Originally Posted by Cactor View Post
    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 Y = 0.7 X_A + 0.3 X_B and I assume that X_A and X_B are independent. Then 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.
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