Consider a random variable X which follows Binomial distribution with parameters n = 10 and . Then Y = 10 - X follows Binomial distribution with parameters n' and p' respectively given by

Concept:

Binomial distribution:

The binomial distribution for P(X = r) with parameters n and p is given as follows:

Where q is the complementary event.

Also, the property of any random variable X,

E(aX + b)  = aE(X) + b

V(aX + b)  = (a²)V(X) 

Calculation:

Given: n = 10 and

Therefore, q = 1 - p 

For random variable X, 

Mean = E(X) = np = 10 × (1/5) = 2

Variance = V(X) = npq = 10 × (1/5) ×(4/5)  = 8/5

Since, Y = 10 - X

⇒ E(Y) = E(10 - X)

⇒ E(Y) = 10 - E(X) = 10 - 2

⇒ 8 = n'p'      ....(1)

Also, V(Y) = V(10 - X) = 0 + (-1)² V(X)

⇒ 8/5 = n'p'q'      ....(2)

Solving 1 and 2, we will get n', p', and q' which will be,

⇒ 8/5 = 8'q'   

⇒ q' = 1/5   

⇒ p' = 4/5   

Now,

⇒ 8 = n'  (4/5)

⇒ n' = 10