Question

Normal approximation to the hypergeometric distribution. Let $n, m, k$ be positive integers and suppose that they tend to infinity in such a way that $$ \frac{r}{n+m} \rightarrow t, \quad \frac{n}{n+m} \rightarrow p, \quad \frac{m}{n+m} \rightarrow q, \quad h\{k-r p\} \rightarrow x $$ where $h=1 / \sqrt{(n+m) p q t(1-t)}$. Prove that $$ \binom{n}{k}\binom{m}{r-k} /\binom{n+m}{r} \sim h \mathrm{n}(x) . $$ Hint: Use the normal approximation to the binomial distribution rather than Stirling's formula.

Name: Problem 10, Chapter 7: An Introduction to Probability Theory and Its Applications: Volume 1
Uploaded: 2024-01-13T08:18:57-08:00
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Description: Normal approximation to the hypergeometric distribution. Let n, m, k be positive integers and suppose that they tend to infinity in such a way that (r)/(n+m) →t, (n)/(n+m) →p, (m)/(n+m) →q, h{k-r p} →x where h=1 / √((n+m) p q t(1-t)). Prove that nkmr-k /n+mr ∼h n(x) . Hint: Use the normal approximation to the binomial distribution rather than Stirling's formula.

   Normal approximation to the hypergeometric distribution. Let $n, m, k$ be positive integers and suppose that they tend to infinity in such a way that

$$
\frac{r}{n+m} \rightarrow t, \quad \frac{n}{n+m} \rightarrow p, \quad \frac{m}{n+m} \rightarrow q, \quad h\{k-r p\} \rightarrow x
$$

where $h=1 / \sqrt{(n+m) p q t(1-t)}$. Prove that

$$
\binom{n}{k}\binom{m}{r-k} /\binom{n+m}{r} \sim h \mathrm{n}(x) .
$$


Hint: Use the normal approximation to the binomial distribution rather than Stirling's formula.

An Introduction to Probability Theory and Its Applications: Volume 1

William Feller 1st Edition

Chapter 7, Problem 10

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Normal approximation to the hypergeometric distribution. Let $n, m, k$ be positive integers and suppose that they tend to infinity in such a way that $$ \frac{r}{n+m} \rightarrow t, \quad \frac{n}{n+m} \rightarrow p, \quad \frac{m}{n+m} \rightarrow q, \quad h\{k-r p\} \rightarrow x $$ where $h=1 / \sqrt{(n+m) p q t(1-t)}$. Prove that $$ \binom{n}{k}\binom{m}{r-k} /\binom{n+m}{r} \sim h \mathrm{n}(x) . $$ Hint: Use the normal approximation to the binomial distribution rather than Stirling's formula.