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December 25th, 2015, 06:39 PM  #1 
Member Joined: Apr 2014 From: Birmingham, AL Posts: 36 Thanks: 4  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? 
January 1st, 2016, 05:57 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
?? 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.)

January 3rd, 2016, 05:34 PM  #3 
Member Joined: Apr 2014 From: Birmingham, AL Posts: 36 Thanks: 4 
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((xy)^2/2a^2)*1/(sqrt(2pi)*c)*exp((yb)^2/2c^2) dy I do not know how to do this integral. I have tried but am stumped. 
January 16th, 2016, 08:18 AM  #4 
Member Joined: Apr 2014 From: Birmingham, AL Posts: 36 Thanks: 4 
Var(X) = E[Var(XY)] + Var(E[XY]) = a + c. Don't I feel like an idiot. This is why you should always read your textbook!


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distribution, figure, mixed distribution, normal, normal distribution 
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