Question

The mean and variance of binomial distribution are 6, 4. Then the parameters of the distribution are
a. $12,\dfrac{1}{2}$
b. $9,\dfrac{2}{3}$
c. $10,\dfrac{3}{5}$
d. $18,\dfrac{1}{3}$

Answer 1

Hint: In order to solve this question, we should know about the formula of mean, that is, np and the formula of variance of binomial distribution, that is, np (1-p) where n represents the number of trials and p represents the probability of success in each trial. So, we will first form the equations of mean and variance and then by solving the equations, we will calculate the values of the parameters of distribution, that are n and p.

Complete step-by-step answer:

In this question, we have been given the mean and variance of a binomial distribution and we have been asked to find the parameter of the distribution. To solve this question, we should know about the formulas of mean and variance. And we know that, mean and variance are given as,
Mean = np
Variance = np (1-p)
Where, n represents the number of trials and p represents the probability of success.
Now, we have been given that mean = 6 and variance = 4. So, we can say,
np = 6 …… (i) and np (1-p) = 4 …… (ii)
Now, we will divide equation (ii) by equation (i). So, we will get,
$\dfrac{np\left( 1-p \right)}{np}=\dfrac{4}{6}$
Now, we know that the common terms get cancelled out, so we can write,
$1-p=\dfrac{2}{3}$
We can further write it as,
$\begin{align}
  & p=1-\dfrac{2}{3} \\
 & p=\dfrac{1}{3} \\
\end{align}$
Now, we will put the value of p in equation (i). So, we will get,
$\begin{align}
  & n\times \dfrac{1}{3}=6 \\
 & n=18 \\
\end{align}$
Hence, we can say that, we get the parameters of the binomial distribution, whose mean and variance are 6, 4 as 18, $\dfrac{1}{3}$. Therefore, we can say that option (d) is the correct answer.

Note: While solving this question, we need to remember that mean is calculated by the formula, np and variance is calculated using the formula, np (1-p), where n represents the number of trials and p represents the probability of success. We can also solve this question, by using the options one by one in the formula of mean and variance, and checking which gives the mean as 6 and variance as 4 and that will be the correct answer, which will be 18, $\dfrac{1}{3}$.