Question

Suppose that Y is the Poisson distribution of the number of students a school has. Find the mode by studying the monotonicity and explaining if the mean is a)5.3 and other case b) 6

Answer 1

(a)

For Mean = = 5.3:

The Probability Mass Function of Poisson Distribution is given by:

,

                            for x = 0, 1 2,...

To find the mode of the Poisson Distribution:

For k > 0,

consider the ratio:

Simplifying RHS, we get:

So,

we note that the ratio:

is 1 for k 3.5

and

is 1 for k 3.5

Thus, we note that P(X=k) is monotonically increasing upto [3.5] = Integral part of 3.5 = 3and then it is monotonically decreasing.

Thus, mode is given by:

3

(b)

For Mean = = 6:

The Probability Mass Function of Poisson Distribution is given by:

,

                            for x = 0, 1 2,...

To find the mode of the Poisson Distribution:

For k > 0,

consider the ratio:

Simplifying RHS, we get:

So,

we note that the ratio:

is 1 for k 6

and

is 1 for k 6

Thus, we note that P(X=k) is monotonically increasing upto = 6 and then it is monotonically decreasing.

Thus, mode is given by:

6 or 5