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Normal Distributions

  1. Nov 4, 2008 #1
    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
     
  2. jcsd
  3. Nov 4, 2008 #2

    statdad

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    I'm not sure what you mean by

    [tex]
    P(S) + P(D) = P(S) P(D)
    [/tex]

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

    If

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

    and they are independent, then the sum [tex] S + D [/tex] is normal, with mean

    [tex]
    \mu_S + \mu_D
    [/tex]

    and variance

    [tex]
    \sigma^2_S + \sigma^2_D
    [/tex]

    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' [tex] P(S) P(D) [/tex] you mean the convolution of the distributions, you could go through that work, but it leads you to the same result I quoted above.
     
  4. Nov 5, 2008 #3

    Redbelly98

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    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".
     
  5. Nov 5, 2008 #4

    HallsofIvy

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    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.
     
  6. Nov 5, 2008 #5

    statdad

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    I answered as I did because

    • the OP used [tex] S, D[/tex] 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
     
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