Calculate Posterior p.f. from Poisson Distribution

[Throwback]

Homework Statement

Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean λ is either 1.0 or 1.5, and the prior of λ is the following:

L(1.0)=0.4 and L(1.5)=0.6

If the roll of tape selected at random is found to have 3 defects, what is the posterior p.f. of λ?

Homework Equations

p(x|λ) = [e^(-λ)]*(λ^x)/x!

L(1.0|X=3) α f(x|λ)L(λ)

The Attempt at a Solution

I think this is all I have to do: (1) calculate the probability of X=3 with parameter λ=1 for a poisson distribution and then multiply this by 0.4. Then do the same thing for λ=1.5 and a prior of 0.6. Right?

So .4*(1/e)*(1/3!)=0.0245253 and .6*(e^(-1.5))*(1.5^3)/3!=0.0753064

According to the answers from the book, this is wrong.

I know the likelihood for the poisson distribution above is

e^(-nμ)*μ^(Ʃx)/(x1!*...*xn!)

but I don't think I have to calculate this the same way I'd calculate the posterior using a gamma distribution, for example.

Any help would be great.

 

[KataKoniK]

Thank you for your question. Your approach to calculating the posterior probability of λ is correct. However, there are a few errors in your calculations.

Firstly, the likelihood function for a Poisson distribution is:

p(x|λ) = (e^(-λ))*(λ^x)/x!

Therefore, for λ=1, the probability of X=3 is:

p(x=3|λ=1) = (e^(-1))*(1^3)/3! = 0.0613132

Similarly, for λ=1.5, the probability of X=3 is:

p(x=3|λ=1.5) = (e^(-1.5))*(1.5^3)/3! = 0.1255107

Now, to calculate the posterior probability, we multiply these probabilities by their respective prior probabilities, as you correctly stated. Therefore, the posterior probability for λ=1 is:

P(λ=1|X=3) = p(x=3|λ=1)*L(λ=1) = 0.0613132*0.4 = 0.0245253

And the posterior probability for λ=1.5 is:

P(λ=1.5|X=3) = p(x=3|λ=1.5)*L(λ=1.5) = 0.1255107*0.6 = 0.0753064

So, the posterior probability distribution for λ is:

P(λ=1|X=3) = 0.0245253

P(λ=1.5|X=3) = 0.0753064

I hope this helps. Please let me know if you have any further questions or if you need any clarification.