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I roll a die until it comes up $6$ and add up the numbers of spots I see.

For example, I roll $4,1,3,5,6$ and record the number $4+1+3+5+6=19$. Call this sum $S$.

Find the standard deviation of $S$.


I have been looking for an easy way to do this because I know I can use the definitions here to calculate the variance of $S$ and then take the square root of it. But I am sure there is an easier way to do this.

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A geometric distribution seems suited to this sort of problem. –  smackcrane Nov 30 '11 at 5:34
    
There is no easier way than calculating the variance first. –  Henry Nov 30 '11 at 7:34
    
@Henry but how to calculate the variance here, it seems Didier has something pretty complicated. –  geraldgreen Nov 30 '11 at 7:54
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3 Answers 3

up vote 8 down vote accepted

I know I can use the definitions here to calculate the variance of $S$ and then take the square root of it.

Not sure I understand what you mean by that... but here we go.

Let $i=6$. For every $n\geqslant1$, call $X_n$ the result of the $n$th throw, uniformly distributed on $\{1,2,\ldots,i\}$, and $A_n$ the event that $X_k\ne i$ for every $1\leqslant k\leqslant n-1$. Then $$ S=\sum\limits_{n=1}^{+\infty}X_n\cdot[A_n]. $$ For every $n$, $\mathrm E(X_n)=x$ with $x=\frac12(i+1)$ and $\mathrm P(A_n)=a^{n-1}$ with $a=\mathrm P(X_1\ne i)$ hence $a=(i-1)/i$ and $$ \mathrm E(S)=\sum\limits_{n=1}^{+\infty}x\cdot a^{n-1}=x\cdot(1-a)^{-1}=\tfrac12i(i+1). $$ Likewise $\mathrm E(S^2)=u+2v$ with $$ u=\sum\limits_{n=1}^{+\infty}\mathrm E(X_n^2\cdot[A_n]), \qquad v=\sum\limits_{n=1}^{+\infty}\sum\limits_{k=n+1}^{+\infty}\mathrm E(X_nX_k\cdot[A_k]). $$ For every $n$, $\mathrm E(X_n^2)=y$ with $y=\mathrm E(X_1^2)=\frac16(i+1)(2i+1)$, and $X_n$ and $A_n$ are independent, hence $$ u=\sum\limits_{n=1}^{+\infty}y\cdot a^{n-1}=y\cdot(1-a)^{-1}=yi. $$ Likewise, for every $k\gt n$, $X_k$ is independent on $X_n\cdot[A_k]$ and $$ \mathrm E(X_n\mid A_k)=\mathrm E(X_n\mid X_n\ne i)=z, $$ with $z=\mathrm E(X_1\mid X_1\ne i)=\frac12i$, hence $$ v=\sum\limits_{n=1}^{+\infty}\sum\limits_{k=n+1}^{+\infty}xz\cdot a^{k-1}=\sum\limits_{n=1}^{+\infty}xz\cdot a^{n}\cdot(1-a)^{-1}=xz\cdot a\cdot(1-a)^{-2}=xzi(i-1). $$ Finally, $$ \mbox{Var}(S)=u+2v-\mathrm E(S)^2=yi+2xzi(i-1)-x^2i^2=\tfrac1{12}i(i+1)(i-1)(3i-2). $$ For $i=6$, $\mathrm E(S)=21$ and $\mbox{Var}(S)=280$.

Edit One sees that $\mathrm E(S)=\mathrm E(X_1)\mathrm E(N)$ where $N$ is the time of the first occurrence of $i$. This is Wald's formula. According to this WP page, the formula for the variance is known as Blackwell–Girshick equation. Proceeding as above, one gets $$ \mathrm{Var}(S)=\mathrm{Var}(X_1)\cdot\mathrm E(N)+\mathrm E(X_1)^2\cdot\mathrm E(N). $$

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Thanks a lot for Wald's formula and Blackwell-Girshick equation reference. –  geraldgreen Nov 30 '11 at 19:56
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Let $Y$ be the number of rolls before a $6$ is rolled. Let $X$ be the sum of the dice rolled before a $6$. Straightforward calculation yields $$ \mathsf{E}(X|Y=n)=n\;\mathsf{E}(X|Y=1)=3n $$ and $$ \mathsf{Var}(X|Y=n)=n\;\mathsf{Var}(X|Y=1)=2n $$ and $$ \mathsf{P}(Y=n)=\left(\frac{5}{6}\right)^n\frac{1}{6} $$ Using the Law of Total Variance, we get $$ \begin{align} \mathsf{Var}(X) &=\mathsf{E}(\mathsf{Var}(X|Y))+\mathsf{Var}(\mathsf{E}(X|Y))\\ &=\sum_{n=0}^\infty\;2n\left(\frac{5}{6}\right)^n\frac{1}{6}+\sum_{n=0}^\infty\;(3n)^2\left(\frac{5}{6}\right)^n\frac{1}{6}-\left(\sum_{n=0}^\infty\;3n\left(\frac{5}{6}\right)^n\frac{1}{6}\right)^2\\ &=\frac{2}{6}\frac{\frac{5}{6}}{(1-\frac{5}{6})^2}+\frac{3^2}{6}\left(\frac{2\left(\frac{5}{6}\right)^2}{(1-\frac{5}{6})^3}+\frac{\frac{5}{6}}{(1-\frac{5}{6})^2}\right)-\left(\frac{3}{6}\frac{\frac{5}{6}}{(1-\frac{5}{6})^2}\right)^2\\ &=10+495-225\\ &=280 \end{align} $$ Therefore, the standard deviation is $\sqrt{280}$.

Afterword:

Although not requested in the question, the expected value of $S$ is simple to compute by the linearity of expectation. Since the probability of rolling a $6$ is $\frac{1}{6}$, the mean number of rolls is $6$. Since each non-$6$ roll has a mean of $3$ and on average there will be $5$ non-$6$ rolls, we get $\mathsf{E}(S)=5\cdot3+6=21$.

We can also compute this using the set-up for the variance above. Since $S=6+X$, $$ \begin{align} \mathsf{E}(S) &=6+\mathsf{E}(X)\\ &=6+\mathsf{E}(\mathsf{E}(X|Y))\\ &=6+\sum_{n=0}^\infty3n\left(\frac{5}{6}\right)^n\frac{1}{6}\\ &=6+\frac{3}{6}\frac{\frac{5}{6}}{(1-\frac{5}{6})^2}\\ &=6+15\\ &=21 \end{align} $$

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A considerably less general and less detailed solution than the one provided by Didier Piau is as follows.

If the first $6$ occurs on the $N$-th roll of the die, then $N$ is a geometric random variable with parameter $\frac{1}{6}$. We have that $E[N] = 6$ and $\text{var}(N) = 30$. Given the value of $N$, we can write $$S = 6 + \sum_{i=1}^{N-1} Y_i$$ where the $Y_i$ are independent random variables uniformly distributed on $\{1,2,3,4,5\}$ and the $6$ is the contribution of the $N$th roll of the die to the sum. Since $E[Y_i] = \frac{1+2+3+4+5}{5} = 3$, $$E[S\mid N] = E\left[6 + \sum_{i=1}^{N-1} Y_i\right] = 6 + \sum_{i=1}^{N-1}E[Y_i] = 6 + 3(N-1) = 3N + 3$$ and so $$E[S] = E[E[S|N]] = E[3N + 3] = 21.$$

Since each $Y_i$ has variance $\dfrac{1^2+2^2+3^2+4^2+5^2}{5}-3^2 = 2$ and is independent of the others, $$\text{var}(S\mid N) = \text{var}\left(6 + \sum_{i=1}^{N-1} Y_i\right) = \sum_{i=1}^{N-1}\text{var}(Y_i) = 2(N-1) ~$$ and so $$E[\text{var}(S\mid N)] = E[2(N-1)] = 10$$ while $$\text{var}(E[S\mid N]) = \text{var}(3N + 3) = 9\cdot\text{var}(N) = 270$$ giving $$\text{var}(S) = E[\text{var}(S\mid N)] + \text{var}(E[S\mid N]) = 280.$$

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Thanks for your recent flag. I just wanted to mention that now that the question has an upvoted answer, it shouldn't be bumped anymore. –  Zev Chonoles Apr 3 '12 at 16:38
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