ML estimation
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.)
unanswered
Answer = k
The maximum likelihood (ML) estimate of λ based on a single observation K=k can be found by maximizing the likelihood function.
The likelihood function is defined as the probability of observing the given data, assuming a particular value for the parameter. In this case, it is the probability of observing K=k given the parameter λ.
The likelihood function L(λ|k) is given by:
L(λ|k) = pK(k;λ) = λ^k * e^(-λ) / k!
To find the ML estimate of λ, we need to find the value of λ that maximizes this likelihood function. This can be done by taking the derivative of the likelihood function with respect to λ and setting it equal to zero.
∂L(λ|k) / ∂λ = 0
Differentiating the likelihood function with respect to λ, we get:
∂L(λ|k) / ∂λ = kλ^(k-1) * e^(-λ) / k! - λ^k * e^(-λ) / k! = 0
Simplifying further, we can cancel out the common terms:
kλ^(k-1) - λ^k = 0
Factoring out λ^(k-1), we get:
λ^(k-1) * (k - λ) = 0
To satisfy this equation, either λ^(k-1) = 0 or (k - λ) = 0. Since λ cannot be zero, we have:
k - λ = 0
Solving for λ, we find:
λ = k
Therefore, the ML estimate of λ based on a single observation K=k is λ = k.
To find the maximum likelihood (ML) estimate of λ based on a single observation K=k, we need to maximize the likelihood function L(λ), which is the probability mass function (PMF) of the Poisson distribution.
The likelihood function L(λ) is given by:
L(λ) = pK(k; λ) = λ^k * e^(-λ) / k!
To find the maximum, we can take the derivative with respect to λ and set it equal to zero:
dL(λ)/dλ = 0
To simplify the calculation, let's take the natural logarithm of the likelihood function and find the maximum of the log-likelihood instead:
ln L(λ) = ln(λ^k * e^(-λ) / k!) = k * ln(λ) - λ - ln(k!)
Taking the derivative of ln L(λ) with respect to λ:
d(ln L(λ))/dλ = k/λ - 1
Setting the derivative equal to zero and solving for λ:
k/λ - 1 = 0
k = λ
Therefore, the ML estimate of λ based on a single observation K=k is λ = k.