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December 25th, 2015, 06:39 PM   #1
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Smile Need some help figure out this normal distribution

I am stumped and need someone better at this than me

I am trying to find the mean and variance of the following normal random variable:

X ~ N(d+eY,a)
Y ~ N(b,c)

a,b,c,d,e are known constants. I would like to solve for the mean and variance of X. I guess the mean would be d+eb. How could I find the variance?
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January 1st, 2016, 05:57 AM   #2
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?? The notation "$\displaystyle N(\mu, \sigma)$" means "the normal distribution with mean $\displaystyle \mu$ and standard distribution $\displaystyle \sigma$. "N(d+ ey, a)" says that the mean is d+ ey and that the standard deviation is a so the variance is $\displaystyle a^2$. (The variance is defined as the square of the standard deviation.)
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January 3rd, 2016, 05:34 PM   #3
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Hey Country Boy, thanks for the reply and we live in the same state!

Yes I do know what this notation means. Maybe I am still not posing the question correctly.

I want to know the unconditional distribution of X. X's mean is a linear combination of the random variable Y.

I wish I knew how to write formulas..

I know the pdf of X is the integral from -infinity to infinity of 1/(sqrt(2pi)*a)*exp(-(x-y)^2/2a^2)*1/(sqrt(2pi)*c)*exp(-(y-b)^2/2c^2) dy

I do not know how to do this integral. I have tried but am stumped.
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January 16th, 2016, 08:18 AM   #4
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Var(X) = E[Var(X|Y)] + Var(E[X|Y]) = a + c. Don't I feel like an idiot. This is why you should always read your textbook!
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