Hey guys, in this video I'm going to do this statistics problem. I haven't made a
youtube video in a while so I at least want to put something on the channel to keep it alive. This is
a maximum likelihood example problem where we're going to be calculating the MLE from a given pdf
and then the second part of the problem, we're given four observations from a sample
and then given our sample, calculating the maximum likelihood estimator for a parameter called tau.
I've done another maximum likelihood estimate video before using a different distribution,
so I'll put a link to that video in the description box below but let's go ahead
and get started with this problem. We're given a pdf and we're asked to estimate tau,
the parameter tau, in the pdf. For calculating the maximum likelihood estimate or MLE, we're
basically going to do the following steps, which are always the same for these types of problems.
We're going to take the product of the pdf and this is going to be our likelihood function.
Then to make it easier to calculate the solution by hand, we're going to take the natural log of
the likelihood function and then we're going to take the partial derivative of the log likelihood
with respect to the parameter that we're trying to solve for. In that case, it's going to be the
partial derivative of the log likelihood over d tau and then we set that equal to zero and try
to solve for tau. That's going to be some equation and that will be our maximum likelihood estimator.
Let's go ahead and do that for this problem. We're given the pdf here, so I'm going to first
take the product of the pdf. That's going to be the product of tau e to the negative
tau x plus 3 tau. When I take the product, I just say that's to the n. So this would be tau times
tau times tau n times or tau to the n. Then this is going to be e to the sum of all these terms.
So the sum of negative tau x plus three tau. I'll write it like that for this step. Next to make it
easier to calculate by hand, we're going to take the natural log of the likelihood. So we're going
to take the natural log of tau n e to the sum of negative tau x plus 3 tau. When you multiply
terms and you take the natural log, that's equivalent to taking the sum of the natural
logs. That's going to be equal to natural log of tau n plus the natural log of e to all of those
terms. And then from here we can simplify. The natural log of tau to the n. I can take this n
out. That's going to be equal to n ln tau plus the natural log of e to something is just going
to be equal to the exponential terms. That's equal to the sum of negative tau x plus three tau. Next
I'm going to take the partial derivative of the log likelihood with respect to tau.
For this, this is going to be equal to the derivative of the natural log of tau,
which would be 1 over tau. And then if we take the partial derivative of this term
with respect to tau, these values will go away. That will be equal to negative the sum of x i's
plus the sum of threes. And then we're going to set that equal to zero and solve for tau. To do that,
I'm going to move the sum terms to the right hand side. That's going to be n over tau is equal to
the sum of x i minus the sum of three and then I'm going to multiply by tau and then divide
by this right hand term. That's going to give me n over the sum of x i minus 3 is equal to
tau and this is our solution. This is going to be the maximum likelihood estimate
for tau. That's the solution for 1 and then for 2, we just have to plug in four observations
to get an actual value of tau hat. So tau hat is going to be equal to n over x i minus 3 and
then you just do that for every observation. For this example problem, we have four observations
and then the values are here. So it's going to be tau hat is equal to 4 over each observation
minus three. It's going to be 3.15 minus 3, 3.65
minus three, et cetera. So it's going to be .15 plus .65 plus .1 plus
.3 on the denominator. I'm just going to write that as 0.15 plus 0.65 plus 0.1 plus 0.3
and that would give me 4 over 1.2 or 3.3 repeating. And we're done with this problem.
*chiptune music*