The picture on the right is the solution to question 5.

For question 5, what is meant by "kernel" and how did the solution work out the posterior without calculating the denominator?

5. A random variable X has a Poisson distribution with mean 2, which is initially 5. The prior distribution of 2 is x 4, which is the same as Gamma(2, 1/2). The pdf assumed to have a chi square distribution with 4 degrees of freedom. What is the of the prior distribution is: posterior distribution of 2 after observing a single value x? f ( 2 ) = ( 0.5 ) 2 ne -0.52 was The likelihood function for a single observation x from a Poisson(2) distribution is: co m 6. Suppose that for any given value of 1, Y has an exponential distribution with f(x / 2) = 2*e-7/*!CC mean 1/1. Also, the probabilities that A equals 1, 2 and 3 are 1/4, 1/2 and 1/4, eHere respectively. So the pdf of the posterior distribution of A is: irs Find the posterior distribution of A if the observed value of Y is 3.2. . V f ( 2 / x ) or f ( 2 ) f ( x / 2 ) = (0.5)2ne-0.57 78e-7/x! is study = 2xtle-37/2 2 2 0 . esou This is the kernel of a Gamma(x+2, 3/2) distribution. r Therefore, the posterior distribution of A after observing a single value x is Gamma(x+2, 3/2)