Question

A Poisson distribution has a double mode at x = 1 and x = 2. The probability for x = 1 or for x = 2 of these two value is:
1. 4e^-2
2. e^-2
3. 2e^-2
4. 3e^-2

Answer 1

Correct Answer - Option 1 : 4e^-2

Formula

Poisson distribution = P(X = x) = [ e^-m m^x/x! for x = 0, 1, 2, 3 -----]

                                                          [    0                       ; otherwise]

Calculation

Poisson distribution has 2 modes at x = 1 and x = 2 it means they have same frequency at x = 1 and 2and have same probability

P(x = 1) = λ¹. e^-λ/1!

⇒ P(x = 2) = λ². e^-λ/2!

According to question P(x = 1) = P(x = 2)

λ¹. e^-λ/1! = λ¹. e^-λ/2!

⇒ λ¹. e^-λ = λ². e^-λ/2

⇒ λ = 2

⇒ P(x = 1) = 2¹. e^-2/1!

⇒ 2e^-2

⇒ P(x = 2) = 2². e^-2/2!

⇒ 2.e^-2

⇒  P(x) = P(x = 1) + P(x = 2)

⇒ 2e^-2 + 2.e^-2

∴ P(x) = 4e^-2