Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\operatorname{Var}(X)$ ?

share|cite|improve this question

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 $\sigma^2$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.