# Multiplying a Normal random variable by arbitrary random variable

I'm going through "Probabilistic Machine Learning: An Introduction" by Murphy (2021), and currently trying to do Ex. 3.1 (p. 80), which states:

Let $$X \sim N(0, 1)$$ and $$Y = WX$$, where $$p(W = 1) = p(W =-1) = .5$$. It is clear that X and Y are not independent, since Y is a function of X.

a. Show $$Y \sim N(0, 1)$$

b. Show $$Cov[X, Y] = 0$$ Thus X and Y are uncorrelated but dependent, even though they are Gaussian. Hint: use the definition of covariance:

$$Cov[X,Y] = \mathbb{E}[XY]−\mathbb{E}[X]\mathbb{E}[Y]$$ and $$\mathbb{E}[XY ] = \mathbb{E}[\mathbb{E}[XY |W]]$$

a) Now, conceptually, I get that $$\mathbb{E}[W] = 0$$ and that $$Y = WX$$ should be distributed $$N(0, 1)$$. But this is just because $$W$$ is such a simple distribution, and I would definitely be stuck if it were more complicated. So everything in between is done in my head because "it seems like it should be so", and not because I can derive it.

b) I also get that $$\mathbb{E}[XY]$$ and $$\mathbb{E}[X]\mathbb{E}[Y]$$ should be 0, therefore $$Cov[EX]=0$$, but also only on an abstract basis.

Could someone please guide me through this? This is not a homework question or any assignment, just trying to plough through the book by myself.

Thanks.

• $Y$ is a mixture of two identically distributed symmetric variables and therefore has the same distribution. When those variables have a finite variance, the covariance must be zero by an analogous symmetry argument--no formulas are needed. See "Solution by circular symmetry" at stats.stackexchange.com/a/257919/919 for how one goes about making this kind of argument. – whuber Feb 23 at 13:27
• @whuber Thanks for the explanation and the link. – Zlo Feb 23 at 13:44

The distribution of $$Y$$ can be factorized as $$p_Y(y) = \sum_{w} p_{Y,W}(y,w) = \sum_{w}p_W(w)p_{Y\mid W}(y\mid w)$$ If you denote $$P(W=1)=q$$ you obtain that $$Y$$ is distributed as $$p_Y(y) = q\,p_X(x) + (1-q)\,p_X(-x)$$ However in your case you have a distribution which is symmetric about zero so that $$p_X(x) = p_X(-x)$$, therefore you obtain $$p_Y = (q + 1 - q)p_X = p_X$$.
You could make the problem more complex by having $$W$$ take $$K$$ non-zero values and use the same formulas. However you will need to derive the pdf of $$Y\mid W=w$$ which is equal to the pdf of $$wX$$: this is simply $$\frac{1}{\lvert w \rvert}p_X(y/w)$$.
Therefore you have that $$Y$$ is distributed as $$p_Y(y) = \sum_{i=1}^K \dfrac{p(w_i)}{\lvert w_i \rvert}p_X(y/w_i)$$