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  • Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r) / P(x = n–r) is independent of n and r, then p equals A. 1/2 B. 1/3 C. 1/5 D. 1/7

Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r) / P(x = n–r) is independent of n and r, then p equals
A. 1/2
B. 1/3
C. 1/5
D. 1/7

Answers (1)

\begin{aligned} &\text { As we know that in binomial distribution } \mathrm{P}(\mathrm{X}=\mathrm{r})={ }^{n} \mathrm{C}_{r}(\mathrm{p})^{\mathrm{r}}(\mathrm{q})^{\mathrm{n}-\mathrm{r}}\\ &\mathrm{P}(\mathrm{x}=\mathrm{r})=\frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) ! \mathrm{r} !}(\mathrm{p})^{\mathrm{r}}(1-\mathrm{p})^{\mathrm{n}-\mathrm{r}} \text { where } \mathrm{q}=1-\mathrm{p}\\ &\text { Therefore, }\\ &\frac{\mathrm{P}(\mathrm{x}=\mathrm{r})}{\mathrm{P}(\mathrm{x}=\mathrm{n}-\mathrm{r})}=\frac{(\mathrm{p})^{\mathrm{r}}(1-\mathrm{p})^{\mathrm{n}-\mathrm{r}}}{(\mathrm{p})^{\mathrm{n}-\mathrm{r}}(1-\mathrm{p})^{\mathrm{r}}}{}\\ &\text { since, }{ }^{n} \mathrm{C}_{r}={ }^{n} \mathrm{C}_{n-r}\\ &\frac{P(x=r)}{P(x=n-r)}=\left(\frac{1-p}{p}\right)^{n-2 r} \end{aligned}

According to the question, this expression is independent of n and r if

\frac{1-p}{p}=1 \Rightarrow p=\frac{1}{2}

Hence  p=\frac{1}{2}
Option A is correct.

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infoexpert22

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