Maximum Likelihood Estimation (MLE) for a Discrete Random Variable: Suppose
the number of calls X received per hour at a call center follows a Poisson distribution with
unknown mean Ī». We want to use MLE to estimate Ī».
(a) Suppose Ī» > 0 is fixed and X ā¼ Poisson(Ī»). For any k = 0, 1, . . ., what is P(X = k)?
(b) Given n IID copies X1, . . . , Xn of X ā¼ Poisson(Ī»), write down an expression for the like-
lihood function
L(Ī») = P(X1 = k1, X2 = k2, . . . , Xn = kn).
(c) The log likelihood function ā(Ī») is obtained by taking the (natural) log of the likelihood
function L(Ī»). Write down an expression for ā(Ī»).
[Use the properties of log, like ln(ab) = ln(a) + ln(b), to simplify your expression for ā(Ī»).]
(d) Find Ī» that maximizes the log likelihood function ā(Ī»). (This is called the maximum
likelihood estimate of Ī».)
Hint: Take the derivative of ā(Ī») with respect to Ī», set it equal to zero and solve for Ī».
(e) Suppose you record the number of calls received between 6-7 pm on ten separate days as
32, 21, 35, 32, 44, 27, 30, 28, 37, 39.
Based on this data, what is the maximum likelihood estimate ĖĪ» for Ī»?
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