# How do I analytically calculate variance of a recursive random variable?

Suppose I have a chest. When you open the chest, there is a 60% chance of getting a prize and a 40% chance of getting 2 more chests. Let $$X$$ be the number of prizes you get. What is its variance?

Computing $$E[X]$$ is fairly straight forward: $$E[X] = .4 \cdot 2 \cdot E[X] + .6$$ which leads to $$E[X] = 3$$, but I'd also like to know the variance of the number of prizes, not just the average. $$Var[X] = E[X^2] - E[X]^2 = E[X^2] - 9$$, but I'm having trouble with $$E[X^2]$$. Anyone have any idea if this is simple? From simulation, I know that the variance is ~30.

Thanks

Call the next chests as $$X_1,X_2$$. With $$0.4$$ probability, our new variable is $$X_1+X_2$$ and with $$0.6$$ probability, it is $$1$$. So,
\begin{align}E[X^2]&=0.4\times E[(X_1+X_2)^2]+0.6\times1^2\\&=0.4\times E[X_1^2+X_2^2+2X_1X_2]+0.6\\&=0.4\times(2E[X^2]+2E[X]^2)+0.6\\&=0.8\times E[X^2]+7.8\rightarrow E[X^2]=39\rightarrow\operatorname{var}(X)=30\end{align}

Actually, it's relatively simple to obtain formulas for the entire distribution as well as an easy procedure to compute any moment of it.

For $$n=1,2,3,\ldots,$$ let $$f_n(p) = \Pr(X=n)$$ with $$p=0.6.$$ Define

$$F_p(t) = f_1(p)t + f_2(p)t^2 + \cdots + f_n(p)t^n + \cdots$$

(the probability generating function). The problem asserts

$$F_p(t) = p\,t + (1-p)F_p^2(t),$$

$$F_p(t) = \frac{1}{2(1-p)}\left(1 \pm \sqrt{1 - 4p(1-p)t}\right).$$

Only the solution with a minus sign makes sense (because the other yields a negative value for $$f_2(p)$$). Expanding it as a formal power series in $$t$$ (using, for instance, the Binomial Theorem) gives

$$F_p(t) = \frac{1}{2(1-p)} \sum_{n=1}^\infty (-1)^{n-1} \binom{1/2}{n} \left(4p(1-p)\right)^n\,t^n = \sum_{n=1}^\infty f_n(p)\,t^n,$$

from which we can read off the entire distribution of $$X$$ term by term. Here's a plot of the log probabilities up to $$n=80$$ created using this formula: (In R:

f <- function(p=0.6, n=1:80) (-1)^(n-1) * choose(1/2, n) * (4*p*(1-p))^n / (2*(1-p))
plot(f(), type="h", log="y")


)

Moreover,

$$E[X] = \frac{\mathrm{d}}{\mathrm{d}t}F_p(t)\bigg|_{t=1} = \frac{p}{\sqrt{1-4p(1-p)}} = 3$$

and

$$E[X(X-1)] = \frac{\mathrm{d}^2}{\mathrm{d}t^2}F_p(t)\bigg|_{t=1} = \frac{2p^2(1-p)}{\sqrt{\left(1 - 4p(1-p)\right)^3}} = 36,$$

whence

$$\operatorname{Var}(X) = E[X(X-1)] + E[X] - E[X]^2 = 36 + 3 - 3^2 = 30.$$

• Wonderful answer! (Wish I could give it more than +1.) For those of us lesser mortals who find it awkward to read binomial coefficients with fractions in them, the mass function can be further "simplified" to: $$f_n(p) = \frac{2}{n} {2(n-1) \choose n-1} p^n (1-p)^n.$$
– Ben
Apr 16, 2020 at 0:56
• (+1) My answer transformed into a pet compared to this one :) Apr 16, 2020 at 9:31
• @Gunes I think your answer is excellent (I upvoted it before even contemplating posting this answer). I created this one because I realized we had few or no posts illustrating techniques of working with pgfs. It was helpful to have your answer available as a check that I was doing the algebra correctly, too.
– whuber
Apr 16, 2020 at 12:02
• @whuber can be also the t script included? Thanks. Apr 17, 2020 at 10:17
• @Maximilian I'm not sure what you are asking for: I posted the R code already.
– whuber
Apr 17, 2020 at 10:54