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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.

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  • $\begingroup$ Is this a homework or practice problem? It sounds like you're already given the variance. $\endgroup$ May 16, 2021 at 15:54
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    $\begingroup$ 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. $\endgroup$
    – jonaden
    May 16, 2021 at 16:00
  • $\begingroup$ Hi, please edit important details like that into the question itself. $\endgroup$ May 16, 2021 at 16:02

1 Answer 1

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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 $\sigma$.

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  • $\begingroup$ @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. $\endgroup$ May 17, 2021 at 7:44
  • $\begingroup$ 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. $\endgroup$
    – Ben
    May 17, 2021 at 9:11
  • $\begingroup$ @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 $\sigma^2$. $\endgroup$ May 17, 2021 at 9:35
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    $\begingroup$ Okay, yes, I see what you mean. That is correct. I have edited the answer to revise this part. $\endgroup$
    – Ben
    May 17, 2021 at 10:42

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