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

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

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

If

$$\begin{align*} S & \sim n(\mu_S, \sigma^2_S)\\ D & \sim n(\mu_D, \sigma^2_D) \end{align*}$$

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

$$\mu_S + \mu_D$$

and variance

$$\sigma^2_S + \sigma^2_D$$

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