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What is the distribution of a random variable that is the product of the two normal random variables ?

Let $X\sim N(\mu_1,\sigma_1), Y\sim N(\mu_2,\sigma_2)$ and $Z=XY$

That is, what is its probability density function, its expected value, and its variance ?

I'm kind of stuck and I can't find a satisfying answer on the web. If anybody knows the answer, or a reference or link, I would be really thankful...

  • 0
    Please check if this [post](http://math.stackexchange.com/questions/133938/what-is-the-density-of-the-product-of-k-i-i-d-normal-random-variables) answers your questions.2012-06-22
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    Saying that each is normally distributed falls short of saying what the joint distribution is. Often it's intended that they are independent but that doesn't get mentioned. But it should be.2012-06-22
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    The distribution is fairly messy. For independent normals with mean $0$, we are dealing with the *product normal*, which has been studied. For general independent normals, mean and variance of the product are not hard to compute from general properties of expectation.2012-06-22
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    @Clara, I attempted to edit your post to make it more clear. Please let me know if anything was incorrect.2012-06-22

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