Variance of production two standard normal distirbution.

Question

Say I have $X \sim \mathcal N(0, 1)$ and $Y\sim \mathcal N(0, 1)$. Is variance of $XY$ must be $1$?

I've use Python do a simulation test:

import numpy as np

mean, var = 0., 0.
for i in range(1000000):
    x = np.random.randn()
    a = np.random.randn()
    z = np.random.randn()
    m = np.random.randn()
    y = x * a * z * m
    mean += y
    var += y**2
mean/1000000, math.sqrt(var/1000000)

The result is (0.0011857316895149386,1.0059557917785764), which saying the Variance is $1$.

The simulation code use four standard normal distribution, I've did production of two got same result.

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Assuming $X$ and $Y$ are independent.

Answer 1

If $X$ and $Y$ are independent (you did not say that in your question) then: $$\mathbb EXY=\mathbb EX\mathbb EY=0$$ so that: $$\mathsf{Var}(XY)=\mathbb EX^2Y^2=\mathbb EX^2\mathbb EY^2=1$$