# Variance of Bernoulli when success probability varies

Say the success probability $$X$$ is a random variable with mean $$\mu$$ and Variance $$\sigma^2$$ which takes values in $$[0,1]$$. How can I compute the variance of a random Variable $$Y$$ which is 1 with probability $$X$$ and 0 with probability $$1-X$$. So there are "two layers" of variance.

• Is this a homework or practice problem? It sounds like you're already given the variance. – Arya McCarthy May 16 at 15:54
• It is just something that I want to understand myself. I mean that there is now another random variable $Y$ which is 1 with probability $X$ (which is random itself) and 0 with probability $1- X$. So there are "two layers" of variance and I dont know how to deal with this. – jonaden May 16 at 16:00
• Hi, please edit important details like that into the question itself. – Arya McCarthy May 16 at 16:02

In general, you solve problems like this using the 'law of iterated variance'.

Let $$Y|X \sim \text{Bern}(X)$$ and use your stipulated prior mean and variance for the success probability $$X$$. Using the law of iterated variance, you get:

\begin{align} \mathbb{V}(Y) &= \mathbb{V}(\mathbb{E}(Y|X)) + \mathbb{E}(\mathbb{V}(Y|X)) \\[6pt] &= \mathbb{V}(X) + \mathbb{E}(X(1-X)) \\[6pt] &= \mathbb{V}(X) + \mathbb{E}(X) - \mathbb{E}(X^2) \\[6pt] &= \mathbb{E}(X^2) - \mathbb{E}(X)^2 + \mathbb{E}(X) - \mathbb{E}(X^2) \\[6pt] &= \mathbb{E}(X) - \mathbb{E}(X)^2 \\[6pt] &= \mu(1-\mu). \\[6pt] \end{align}

As you can see, the variance of $$Y$$ is determined by $$\mu$$, and is unaffected by $$\sigma$$. It turns out that this is true for all the moments of $$Y$$. In fact, we have $$\mathbb{E}(f(Y)) = f(0) + [f(1)-f(0)] \mu$$, so the expectation of any function of $$Y$$ is determined by $$\mu$$ and is unaffected by $$\sigma$$.

• Thank you. This answers my question perfectly! However, I think there is a slight mistake in the second to last row, which also changes the result. – jonaden May 16 at 23:19
• @Ben Surely, higher-order moments of $Y$ are not affected by $\sigma^2$ when $Y|X\sim \operatorname{Bern}(X)$ since the marginal point mass function of $Y$ can not be anything other than that of a Bernoulli random variable. – Jarle Tufto May 17 at 7:44
• The higher-order moments still depend on $X$, and $\sigma$ affects the distribution of $X$, so I would expect it to affect the higher-order moments. – Ben May 17 at 9:11
• @Ben No, the marginal distribution of $Y$ is clearly a Bernoulli distribution. Once its mean $EY=E(EY|X)=EX=\mu$ is given, the two point masses can only be $P(Y=0)=1-\mu$ and $P(Y=1)=\mu$ and the higher order moments are the same as that of any ordinary Bernoulli distribution and do not depend on $\sigma^2$. – Jarle Tufto May 17 at 9:35
• Okay, yes, I see what you mean. That is correct. I have edited the answer to revise this part. – Ben May 17 at 10:42