Poisson distribution question?

Poisson Distribution Questions

The procedure indicated can be followed if scientific calculator is NOT available

Mean = lambda = m

According to Poisson distribution

P(x) = m^x / [e^m * x!]

e = 2.71828

P(x=0) = m^0 /[e^m * 0!] = 0.09

1 / [e^m * 1] = 0.09

1/e^m = 0.09

0.09*e^m = 1

e^m = 1/0.09 = 11.11

2.71828^m = 11.11

TRIAL and ERROR method is used to find the value of m

If m = 2

2.71828^2 < 11.11

Because Left Hand Side = 7.39 -----> therefore m=2 is NOT correct

If m=3

2.71828^3 > 11.11

Because Left Hand Side = 20.09 -----> therefore m=3 is NOT correct

m value falls between 2 and 3

If m=2.4

2.71828^2.4 = 11.02 which is close to 11.11

If m=2.405

2.71828^2.405 = 11.08 which is very close to 11.11

If m = 2.408

2.71828^2.408 = 11.11

on rounding to 2 dps m = 2.41 Precise value can be obtained

a) Mean = lambda = m = 2.41

b) Variance = lambda = m = 2.41

Standard deviation = sqrt Variance = sqrt 2.41

The formula that applies here is

P(n) = µ^n / e^µ n!. So here, n = 0

and P(0) = 0.09. Plugging in,

µ^0 / e^µ 0! = 0.09.

1 / e^µ = e^-µ = 0.09

- µ = ln 0.09 ≈ - 2.408

so µ ≈ 2.41, to 2 places