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A coin with probability of getting head $0.6$ is tossed repeatedly till two heads appear. Let $X$ be the number of tosses needed to get exactly 2 heads. Describe the sample space. Find the mean and variance of $X$.

What I've tried so far:

Let $P(H)=p$ and $P(T)=q$ such that $p+q=1$. $\displaystyle P(X=n)=\binom{n}{2}p^2q^{n-2}\tag*{}$ Is this even correct because $\sum\limits_{n} P(X=n)=\frac{p^2}{(1-q)^3}>1$?

I am absolute beginner. Any help would be appreciated.

Update: Reading about negative binomial distribution, I realised the point I was missing. It's that we have to necessarily get the second head on the last toss. The correct P.M.F. would be: $\displaystyle\boxed{ P(X=n)=\binom{n-1}{1}p^2q^{n-2}}\tag*{}$

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    $\begingroup$ Take a look at the negative binomial distribution. $\endgroup$
    – jblood94
    Feb 15, 2023 at 17:14
  • $\begingroup$ @jblood94 Thank you! I've understood my mistakes. Everything now makes sense. $\displaystyle P(X=n)=\binom{n-1}{1}p^2q^{n-2}\tag*{}$ Also, $\sum\limits_{n\geqslant 2}P(X=n)=\frac{p^2}{(1-q)^2}=1\tag*{}$ $\endgroup$ Feb 15, 2023 at 19:45
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    $\begingroup$ I posted my answer as you were posting your update. $\endgroup$
    – Mkanders
    Feb 15, 2023 at 20:18

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I see you learned that your formula was incorrect because you know the last (nth) trial must be a success (heads). That means you have $n-1$ trials to get $r-1$ successes and $n-r$ failures. $$P(n;r) = p\binom{n-1}{r-1}p^{r-1}q^{n-r}$$ In your case, $r=2$, so $$P(n;2) = p\binom{n-1}{1}pq^{n-2}$$

Did you figure out the mean and variance? Here's what I did.

The First Success distribution is the number of trials $Y$ until the first success. $$P(Y=k) = q^{k-1}p$$ $$E(Y) = \sum_{k=1}^{\infty}kq^{k-1}p = \frac{1}{p}$$ $$Var(Y) = \frac{q}{p^2}$$ To get the variance, first get the second moment: $$E(Y^2) = \sum_{k=1}^{\infty}k^2q^{k-1}p = \frac{1+q}{p^2}$$ Then use $$Var(Y) = E(Y^2) -(EY)^2 =\frac{1+q}{p^2} - \frac{1}{p^2} = \frac{q}{p^2}$$ Consider the number of trials until success $r$. This is the sum of $r$ independent variables $Y_1, Y_2, ..., Y_r$ $$E(Y_1 + ... + Y_r) = E(Y_1) + ... +E(Y_r) = r \cdot \frac{1}{p} = \frac{r}{p}$$ Since they're independent, you can add their variances. $$Var(Y_1 + ... + Y_r) = r \cdot \frac{q}{p^2}$$ In your case, $r=2, p = 0.6, q = 0.4$. $$E(Y_1 + Y_2) = \frac{2}{0.6} = \frac{10}{3} = 3.333$$ $$Var(Y_1 + Y_2) = 2 \cdot \frac{0.4}{0.6^2} = 2 \cdot \frac{10}{9} = 2.222$$

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  • $\begingroup$ I had managed to get the mean & the variance using $E(X)$ and $E(X^2)$ but it was quite lengthy... This is much easier! Thank you for this interesting approach. $\endgroup$ Feb 15, 2023 at 22:29
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    $\begingroup$ variance should have been 2.222. Correct now. $\endgroup$
    – Mkanders
    Feb 16, 2023 at 4:42

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