00:01
Okay, so to find the mle for lambda, we first need to write down the likelihood function, which is the likelihood function of observing the data that we do, given that lambda is the parameter.
00:15
And this is just the product, since we assume the data is independent and identically distributed.
00:20
It's the product from i equals 1 to n of the probability function for each one, which is lambda to the xi times e to the minus lambda divided by xi factorial.
00:28
And this turns out to be lambda to the sum of the xi's times e to the minus n lambda divided by the product from i equals 1 to n of xi factorial.
00:43
We then take the log likelihood which is the natural logarithm of this and that just gives us the sum of xi times log of lambda minus n lambda lambda minus the sum of the log of xi factorial.
01:07
And then to find the mle of lambda, we differentiate this with respect to lambda and set it to 0.
01:13
And we say the value for which that happens, which i call lambda hat, is the mle.
01:17
So the differential of this with respect to lambda is the sum of the xi's over lambda minus n.
01:23
And if we set that to zero and set lambda equal to lambda hat we find that lambda hat is equal to one over n times the sum of the xi's.
01:37
For part b then we're then asked to calculate the mean squared error for this lambda hat and that is just given by the expectation of lambda minus lambda hat squared which is the expectation of lambda squared minus twice lambda the expectation of lambda hat plus the expectation of lambda hat squared.
02:07
The expectation of lambda squared is just lambda squared.
02:09
The expectation of lambda hat we can see is just 1 over n times the sum of the expectations of the xi since all the xi follow the poisson distribution with parameter lambda their their expectations are lambda.
02:25
And so n of them is just n lambda.
02:30
So we see the expectation of lambda hat is equal to lambda.
02:34
And so this becomes the expectation of lambda hat squared minus lambda squared.
02:43
So now we just need the expectation of lambda hat squared...