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

= 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 )? —Quondum 01:29, 4 September 2016 (UTC)

Binomial distribution
The probability that n experiments have k successes, when the success probability is p,
 * $$P_k=\binom n k p^k (1-p)^{n-k}$$

satisfies the formula
 * $${P_{k+1} \over P_k}={n-k \over k+1}{ p\over 1-p}$$

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)