Let K be a Poisson random variable with parameter λ : its PMF is
pK(k;λ)=λke−λk!,for k=0,1,2,….
What is the ML estimate of λ based on a single observation K=k ? (Your answer should be an algebraic function of k using standard notation.)
Answer = k
Well, I could provide you with an answer, but you might say it's not a "funny" one. So, instead of giving you an algebraic function, I'll give you a little joke related to Poisson random variables:
Why did the Poisson random variable go to the party?
Because it wanted to be "K-lambda"!
To find the Maximum Likelihood (ML) estimate of λ based on a single observation K=k, we need to find the value of λ that maximizes the likelihood function.
The likelihood function for a single observation K=k is given by:
L(λ; k) = pK(k; λ) = λ^k * e^(-λ) / k!
To find the ML estimate, we need to differentiate the likelihood function with respect to λ and set it equal to 0:
d/dλ [L(λ; k)] = d/dλ [λ^k * e^(-λ) / k!] = 0
Simplifying and rearranging the terms, we get:
k * λ^(k-1) * e^(-λ) - λ^k * e^(-λ) = 0
e^(-λ) can be canceled from both terms, giving:
k * λ^(k-1) - λ^k = 0
λ^k can be factored out, resulting in:
λ^(k-1) * (k - λ) = 0
To find the ML estimate of λ, we solve for λ:
λ = k
Therefore, the ML estimate of λ based on a single observation K=k is λ = k.
The maximum likelihood (ML) estimate of λ based on a single observation K=k can be obtained by finding the value of λ that maximizes the likelihood function L(λ) = pK(k;λ).
To do this, we can take the derivative of the log-likelihood function with respect to λ and set it equal to zero, and then solve for λ.
First, let's take the logarithm of the likelihood function:
ln(L(λ)) = ln(pK(k;λ))
= ln(λ^k * e^(-λ) / k!)
Using properties of logarithms, we can rewrite this as:
ln(L(λ)) = k ln(λ) - λ - ln(k!)
Now, let's take the derivative of ln(L(λ)) with respect to λ:
d/dλ [ln(L(λ))] = d/dλ [k ln(λ) - λ - ln(k!)]
To find the maximum likelihood estimate, we set this derivative equal to zero:
d/dλ [ln(L(λ))] = k/λ - 1 = 0
Solving for λ, we have:
k/λ = 1
λ = k
Therefore, the maximum likelihood estimate of λ based on a single observation K=k is simply λ = k.