Testwiki:Reference desk/Archives/Mathematics/2016 September 4

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September 4

Translation of Bourbaki to English

Bourbaki's Algebra chapters 1–3 and 4–7 have been translated to English. Were the remaining chapters translated (most particularly Template:Citation)? —Quondum 01:29, 4 September 2016 (UTC)

Binomial distribution

The probability that n experiments have k successes, when the success probability is p,

$P_k=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$

satisfies the formula

$\frac{P_{k+1}}{P_k}=\frac{n-k}{k+1}\frac{p}{1-p}$

I wonder how the latter formula can be proved without reference to the former formula?

Bo Jacoby (talk) 18:01, 4 September 2016 (UTC).

I have not gone through the details, but it seems promising to me to use the simple (i.e. obvious) rule of recursion

$P_{n,k}=p\cdot P_{n-1,k-1}+(1-p)\cdot P_{n-1,k}$

plus suitable boundary conditions, and then use your formula with induction. By "boundary conditions", I am referring to the cases $k=0$ and $k=n$, which must be proved separately. —Quondum 21:19, 4 September 2016 (UTC)