Question

The Poisson distribution, with unknown parameter λ>0, has probability function

p(x)=eλ(λˣ)/x!
For a random sample X₁, X₂..........., Xₙ, find the maximum likelihood estimator of λ

Answer 1

To find the maximum likelihood estimator of λ for a random sample from a Poisson distribution, we need to find the value of λ that maximizes the likelihood function. The maximum likelihood estimator for λ is the value that maximizes this likelihood function.

Step-by-step explanation:

To find the maximum likelihood estimator of λ for a random sample X₁, X₂, ..., Xₙ from a Poisson distribution, we need to find the value of λ that maximizes the likelihood function. The likelihood function is the product of the probability function of the Poisson distribution for each observation in the sample. The maximum likelihood estimator for λ is the value that maximizes this likelihood function.

To find the maximum likelihood estimator, we can take the derivative of the log-likelihood function with respect to λ, set it equal to 0, and solve for λ.

Let's denote the log-likelihood function as L(λ). Taking the derivative of L(λ) and setting it equal to 0, we get:

dL(λ)/dλ = (∑(Xᵢ) - nλ)/λ = 0

Simplifying, we get:

nλ = ∑(Xᵢ)

Finally, we can solve for λ:

λ = (∑(Xᵢ))/n