First slide

Random variable and probability distribution

Question

Suppose $x$ follows binomial distribution with parameters $n$ and $p$ where $0 < p < 1.$ If $\frac{P (x = r)}{P (x = n - r)}$ is independent of $n$ and $r$ then $p =$

Moderate

A

$\frac{1}{2}$
B

$\frac{1}{3}$
C

$\frac{1}{4}$
D

1

Solution

$\frac{P (X = r)}{P (X = n - r)} = \frac{n_{C_{r}} q^{n . r} p^{r}}{n_{C_{n - r}} q^{r} p^{n - r}}$
= ${(\frac{q}{p})}^{n - 2 r}$

This is independent of n and r if

$\frac{q}{p} = 1 \Rightarrow p = q = \frac{1}{2}$

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