# How to figure out the expectation and variance of this situation

Suppose we randomly draw $n$ balls from an urn with $m$ red balls and $N-m$ white balls. Let $X$ be the number of red balls in our sample. Then $X$ has a hypergeometric distribution with pmf $p(x)=P(X=x)= {_mC_x} \times _{N-m}C_{n-x}/_NC_n$. What is the $E(X)$ and $\operatorname{Var}(X)$ ?

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## 1 Answer

Hint: $X=\sum\limits_{k=1}^nX_k$ where $X_k$ is the indicator function of the event that the $k$th ball is red.

Steps: Show that $x=\mathrm E(X_k)$ does not depend on $k$ and compute $x$. Write $\mu=\mathrm E(X)$ as a function of $x$ and the other parameters of the model. Deduce the value of $\mu$. Show that $\mathrm E(X_k^2)=x$ for every $k$. Show that $y=\mathrm E(X_kX_\ell)$ does not depend on $(k,\ell)$ if $k\ne\ell$ and compute $y$. Write $\sigma^2=\mathrm{Var}(X)$ as a function of $x$, $y$ and the other parameters of the model. Deduce the value of $\sigma^2$.

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I made a trivial edit to work around a formatting bug that I sometimes see, where formulas are placed at the top left of a paragraph instead of at the beginning of the last line. Have you ever seen this? I can still see it when I click on the "edited X mins ago" link -- does your original version appear correctly formatted to you there? – joriki Nov 18 '11 at 7:47
@joriki: No. Yes. (Thanks anyway.) – Did Nov 18 '11 at 7:50
@joriki I've seen this bug in one of my questions before. What is the workaround you suggest? – Chris Taylor Nov 18 '11 at 8:34
@Chris: It's not much of a workaround; I just change the text slightly to move the formula a bit :-) Can you tell me which browser and operating system (including version numbers) you're using? Then I'll post a bug report on meta. – joriki Nov 18 '11 at 9:06
Chrome 15.0.874.106m on Windows 7 64-bit version 6.1 (service pack 1). Although, since Chrome auto-updates itself all the time, I was probably using a different version when I saw the bug... also, I was quite possibly on my laptop instead of my work PC, which runs Mac OSX Leopard 10.5.8. In conclusion, I'm not being very helpful :) – Chris Taylor Nov 18 '11 at 9:22