00:01
Ok, so in a previous question, we showed that the jeffrey prior was given by just one over lambda.
00:09
And so we're going to use that now to answer the following questions.
00:15
So now we want the posterior distribution of lambda given the x's.
00:22
And that is proportional to the jeffrey prior times the likelihood function of the x's given lambda.
00:36
And now this is just equal to one over lambda times lambda to the n e to the minus lambda times the sum of the x's.
00:46
And the proportionality constant.
00:50
So if i'll call that k to account for the proportionalness, we just need to ensure that it integrates to one when we integrate over the possible values of lambda, which from naught to infinity.
01:05
And that is equal to k times gamma n over the sum of the x's to the power n and so we can see that k in order to ensure that this equals 1 we can see that k is equal to the sum of the x's to the power n over gamma n and so we can see that our posterior distribution is equal to lambda to the n minus 1 e to the minus n times the sum of the x's times the sum of the x's to the power n over gamma n and substituting in n equals 10 and the sum of the x's equals 12 we find that this gives us 5, 9, 7, 1, 9, 6, 8 over 35 e to the minus 12 lambda, lambda to the 9...