Question

(a) State three (3) characteristics ofthe binomial distribution.
(b) If X is a binomial random variable following -ben, p),O < p < 1 such that
16P(X = 1) = 4Var(X) = HeX). Find HeX). [8 marks]

Answer 1

There are three characteristics of a binomial experiment:

1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter $n$  denotes the number of trials.

2. There are only two possible outcomes, called success and failure, for each trial. The outcome that we are measuring is defined as a success, while the other outcome is defined as a failure. The letter $p$  denotes the probability of a success on one trial, and $q$  denotes the probability of a failure on one trial. $p+q=1.$

3. The $n$  trials are independent and are repeated using identical conditions. Because the $n$  trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, $p$ , of a success and probability, $q$ , of a failure remain the same.

$$P(X=x)=\dbinom{n}{x}p^xq^{n-x}$$

$$P(X=1)=\dbinom{n}{1}p^1q^{n-1}=npq^{n-1}$$

$$E(X)=np$$

$$Var(X)=\sigma^2=npq$$

$$16P(X=1)=4Var(X)$$

$$16npq^{n-1}=4npq$$

$$q^{n-1}=\dfrac{1}{4}, 0<q<1$$

$$n-1=1, q=\dfrac{1}{4}$$

$$n=2, q=\dfrac{1}{4}$$

Or

$$n-1=2, q=\dfrac{1}{2}$$

$$n=3, q=\dfrac{1}{2}$$

$$4npq=H\cdot E(X)$$

$n=2, q=\dfrac{1}{4}, p=\dfrac{3}{4}$

$$H\cdot E(X)=4(2)(\dfrac{1}{4})(\dfrac{3}{4})=\dfrac{3}{2}$$

$$E(X)=\dfrac{3}{2}, H=1$$

$n=3, q=\dfrac{1}{2}, p=\dfrac{1}{2}$

$$H\cdot E(X)=4(3)(\dfrac{1}{2})(\dfrac{1}{2})=3$$

$$E(X)=\dfrac{3}{2}, H=2$$