Question

6-E4. The distribution below is the binomial distribution with n=4 and p=0.3. You have formulas which allow you to immediately calculate the mean and standard deviation. The point of this exercise is to convince you that the formulas are correct, and also to check that you know what mean and standard deviation are. x Prob 0 0.2401 1 0.4116 2 0.2646 3 0.0756 4 0.0081 What is the mean and standard deviation using the quick formula appropriate for the binomial distribution? Calculate the mean and standard deviation from first principles and hence confirm that the quick formulas give the right answer.

Answer 1

Let X follow Binomial distribution

Probability mass function of x is

with n=4 , p=0.3

Mean of X is

  

= 0*0.2401+1*0.4116+ 2*0.2646 + 3*0.0756 +4 * 0.0081

= 1.2

We know that E(X) = np=4*0.3 = 1.2

= 2.28

V(X) = 2.28-1.2*1.2

= 0.84

We know that, V(X) = np(1-p) = 4 *0.3*(1-0.3) = 0.84

Standard deviation of X =