Multiplying Normal Distributions: Rules & Examples

[chota]

Hi say I have two "independent" Normal distributions,

S ~ N(0,3^2) and D~(0,2^2)

since I know that S and D are indpendent then

P(S ) + P(D) = P(S)P(D)

however we know they are both normal distributed so I amm just wondering what the general rule is for multiplying two normal distributions
thanks

 

[statdad]

I'm not sure what you mean by

<br /> P(S) + P(D) = P(S) P(D)<br />

Are you trying to say that when normal random variables are added, the resulting random variable is their product? Not true.

If

<br /> \begin{align*}<br /> S &amp; \sim n(\mu_S, \sigma^2_S)\\<br /> D &amp; \sim n(\mu_D, \sigma^2_D)<br /> \end{align*}<br />

and they are independent, then the sum S + D is normal, with mean

<br /> \mu_S + \mu_D<br />

and variance

<br /> \sigma^2_S + \sigma^2_D<br />

A similar result is true even if the two variables have non-zero correlation (the formula for the variance of the sum involves the correlation).

If by 'product' P(S) P(D) you mean the convolution of the distributions, you could go through that work, but it leads you to the same result I quoted above.

 

[Redbelly98]

... since I know that S and D are indpendent then

P(S ) + P(D) = P(S)P(D)

I'm guessing you meant to say

P(S & D) = P(S)P(D)

where "S" here really means a statement along the lines of "S lies between A and B", and similarly for "D".

 

[HallsofIvy]

For events A and B, normally distributed or not, P(A&B)= P(A)P(B|A)= P(B)P(B|A) where P(A|B) and P(B|A) are the "conditional probabilities" : P(A|B) is "the probability that A will happen given that B happened" and P(B|A) is "the probability that B will happen given that A happened".

IF the A and B are independent then P(A|B)= P(A) and P(B|A)= P(B) so you just multiply the separate probabilities. If they are not independent, just knowing the probabilities of each separately is not enough. You must know at least one of P(A|B), P(B|A) or P(A&B) separately from the individual probabilities.

 

[statdad]

I answered as I did because

• the OP used S, D in his notation, and I took these as the names of the random variables rather than any interval or event.
• I took the question to mean he was asking how to combine normal distributions rather than calculate any particular probability