# Segment 13 Sanmit Narvekar

## Segment 13

#### To Calculate

1. With p=0.3, and various values of n, how big is the largest discrepancy between the Binomial probability pdf and the approximating Normal pdf? At what value of n does this value become smaller than $\displaystyle 10^{-15}$ ?

2. Show that if four random variables are (together) multinomially distributed, each separately is binomially distributed.

Consider 4 random variables with number of occurrences $\displaystyle A, B, C, D$ and probabilities of occurrence $\displaystyle P_A, P_B, P_C, P_D$ respectively, such that $\displaystyle A + B + C + D = n$ and $\displaystyle P_A + P_B + P_C + P_D = 1$ .

The multinomial distribution over these variables is (with some abuse of notation) given below (think about choosing the A spots for the first random variable, with some probability. Then choosing spots out of the ones remaining for the others, etc..):

$\displaystyle P(A, B, C, D) = \left( \binom{A+B+C+D}{A} (P_A)^A \left( \binom{B+C+D}{B} (P_B)^B \left( \binom{C+D}{C} (P_C)^C \binom{D}{D} (P_D)^D \right) \right) \right)$

Then, by repeatedly applying the binomial theorem:

$\displaystyle P(A, B, C, D) = \left( \binom{A+B+C+D}{A} (P_A)^A \left( \binom{B+(C+D)}{B} (P_B)^B (P_C + P_D)^{C+D} \right) \right)$

$\displaystyle P(A, B, C, D) = \binom{A+(B+C+D)}{A} (P_A)^A (P_B + P_C + P_D)^{B + C+D}$

And using our notation from above

$\displaystyle P(A, B, C, D) = \binom{n}{A} (P_A)^A (1-P_A)^{n-A}$

gives the familiar form of the binomial distribution, where one event is A, and the other event is not A (that is B or C or D). Similar calculations can be done for the others by changing the order in which you select spots for the variables.

1. The segment suggests that $\displaystyle A\ne T$ and $\displaystyle C\ne G$ comes about because genes are randomly distributed on one strand or the other. Could you use the observed discrepancies to estimate, even roughly, the number of genes in the yeast genome? If so, how? If not, why not?
2. Suppose that a Bayesian thinks that the prior probability of the hypothesis that "$\displaystyle P_A=P_T$ " is 0.9, and that the set of all hypotheses that "$\displaystyle P_A\ne P_T$ " have a total prior of 0.1. How might he calculate the odds ratio $\displaystyle \text{Prob}(P_A=P_T)/\text{Prob}(P_A\ne P_T)$ ? Hint: Are there nuisance variables to be marginalized over?