Question

The mode of binomial distribution for which mean and standard deviation are 10 and $\sqrt 5 $ respectively, is
$
  A)7 \\
  B)8 \\
  C)9 \\
  D)10 \\
$

Answer 1

Hint: The mean of binomial distribution is m=np and variance =npq and since we know also that variance is equal to standard deviation .So, by using these values we can find the mode.

Complete step-by-step answer:
Given mean of binomial distribution =10

We know that mean of binomial distribution is =np

So, here np=10

And given that standard deviation = $\sqrt 5 $

We also know that

$
   \Rightarrow {{\text{(variance)}}^2} = ({\text{standard deviation)}} \\

   \Rightarrow {\text{variance}} = \sqrt {{\text{standard deviation}}} \\
$

So, by using the above relation we get

Variance =npq=5

We know that variance =npq and mean =np

Now to calculate q value let us substitute mean value in variance where we get

Here npq=5 where np=10

$
   \Rightarrow 10 \times q = 5 \\

   \Rightarrow q = \dfrac{1}{2} \\

$ (Since np=10)

In binomial distribution generally p is the complement of q.

So, p=q - 1

$
   \Rightarrow p = \dfrac{1}{2} - 1 \\

   \Rightarrow p = \dfrac{1}{2} \\
$

Now here we have to find the mode value .So let us consider that mode as x


To get the mode value, let us make us of condition where

$
   \Rightarrow np + q > x > np - q \\

    \\
$

Let us substitute the values in above condition we get

$
   \Rightarrow 10 + \dfrac{1}{2} > x > 10 - \dfrac{1}{2} \\

   \Rightarrow \dfrac{{21}}{2} > x > \dfrac{{19}}{2} \\

   \Rightarrow 9.5 < x < 10.5 \\
$

From this we can say that x value is equal to 10

Therefore mode of binomial distribution (x) =10

 Option D is the correct answer.

NOTE: In the above problem, we need the variance value but standard deviation value is given. So to get variance value we have used the relation between them where variance is equal to standard deviation. Generally we all ignore this type conversion and solve the problem which has to be avoided.