Distribution of $x_1 x_2+x_1 x_3$ with iid $x_i \sim N(0,1)$

Asked Dec 11 '21 at 00:14

Active Dec 11 '21 at 00:32

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Given are 3 independent standard normal variables $x_1,x_2,x_3$.

What is the pdf of $x_1 x_2+x_1 x_3$?

What is known?

The pdf of the product of two standard normal variables is the Bessel function $K_0(|x|)/\pi$.

Convolution cannot be applied as the summands are dependent.

edited Dec 11 '21 at 00:32

asked Dec 11 '21 at 00:14

user269684

1

Hint: For simplicity, use $X_1(X_2+X_3)$ where the second factor is $\mathsf{Norm}(\mu=0,\sigma=\sqrt{2})$ and factors are independent. – BruceET Dec 11 '21 at 00:37
You can find the answer from https://stats.stackexchange.com/questions/69134/when-is-the-distribution-of-product-of-two-normal-distributed-variables-near-nor – kjetil b halvorsen Dec 11 '21 at 19:04

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