Normalization constant for the Poisson distribution
The probability mass function (pmf) of a Poisson distribution with parameter lambda is given by
Poi(lambda) = c*lambda^k / k!
Where k = 0,1,2,....
Compute the value of c
To compute the value of c, we need to find the normalization constant that ensures that the total probability of all possible outcomes sums up to 1.
The sum of the probabilities for all possible values of k can be represented as:
1 = ∑ [c * lambda^k / k!]
To find the value of c, we need to find the sum of this infinite series. One way to do this is by expressing the exponential term (lambda^k / k!) as the power series expansion for e^lambda:
e^lambda = ∑ [lambda^k / k!]
Multiplying both sides of this equation by c, we get:
c * e^lambda = ∑ [c * lambda^k / k!]
Since the sum on the right side is the same as the sum we are trying to find the value of c for, we can replace it with 1:
c * e^lambda = 1
Therefore, the value of c is equal to the inverse of the exponential function evaluated at lambda:
c = e^(-lambda)
So, the value of c for the Poisson distribution is e^(-lambda).