# Examples of probability distributions that the product of their random variable is also in same distribution.

Heads up: my statistics knowledge is very limited, I am not a mathematician.

I've recently learned that given two random variables $$x$$ and $$y$$ sampled from the same normal distribution their product $$xy$$ is not normal but it instead belongs to a modified bessel function distribution of the second kind (I believe this is only for $$\mu=0$$ and $$\sigma = 1$$).

What are some well known probability distributions such that $$xy$$ belongs to the same distribution? I am specially interested in symmetric distributions around the mean.

Suppose that $$X,Y$$ are random variables and consider $$Z = XY$$ \begin{align} F_Z(z) & = P(Z \le z) \\ & = P(XY \le z) \\ & = P(X \le z/Y | Y > 0)P(Y>0) + P(X \ge z/Y | Y < 0) P(Y<0) \\ & = \int_0^\infty \int_{-\infty}^{z/y} f_{X,Y}(x,y) \ dx dy + \int_{-\infty}^0 \int_{z/y}^\infty f_{X,Y}(x,y) \ dx dy \\ f_Z(z) & = \int_0^\infty f_{X,Y}(z/y,y) \frac{1}{y} \ dy - \int_{-\infty}^0 f_{X,Y}(z/y,y) \frac{1}{y} \ dy \\ f_Z(z) & = \int_{-\infty}^\infty f_{X,Y}(z/y,y)\frac{1}{|y|} \ dy. \end{align} Now suppose that $$X,Y$$ are i.i.d. with common density function $$f$$ such that $$f(x) = f(-x)$$, then the above reduces to $$f_Z(z) = \int_{-\infty}^\infty \frac{f(z/y) f(y)}{|y|} \ dy$$ What you require involves solving the integral equation $$f(z) = \int_{-\infty}^\infty \frac{f(z/y) f(y)}{|y|} \ dy = 2\int_0^\infty \frac{f(z/y)f(y)}{y} \ dy,$$ which to me is not obvious how to do.
If $$X$$ and $$Y$$ are allowed to be discrete random variables rather than only continuous random variables such as normal random variables, then when $$X$$ and $$Y$$ are independent Bernoulli random variables with parameters $$p$$ and $$q$$ respectively, $$XY$$ is also a Bernoulli random variable with albeit with a different parameter $$pq$$.
If you want symmetric distributions, take $$X$$ and $$Y$$ to be independent Bernoulli random variables with parameter $$\frac 12$$. Then, $$W = (-1)^X$$ and $$Z = (-1)^Y$$ are independent random variables taking on values $$\pm 1$$ with equal probability $$\frac 12$$ (i.e. have symmetric distributions). Then, $$WZ$$ also takes on values $$\pm 1$$ with equal probability $$\frac 12$$.