Suppose X is a Poisson random variable with rate parameter A, where A is a random variable itself following a gamma prior distribution with known parameters a and B. (a) Show that the posterior distribution of A given X = z is also a gamma distribu- tion, but with paramaters a +z and 3+1 [Note: we have now shown that the gamma distribution is the conjugate prior for the Poisson distribution.] (b) Show that the Bayesian estimator, given by the mean of the posterior distribution for A, is B(a +1)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![2. Suppose X is a Poisson random variable with rate parameter A, where A is a random
variable itself following a gamma prior distribution with known parameters a and ß.
(a) Show that the posterior distribution of A given X = 1 is also a gamma distribu-
tion, but with paramaters a + z and
3+1
[Note: we have now shown that the gamma distribution is the conjugate prior for
the Poisson distribution.]
(b) Show that the Bayesian estimator, given by the mean of the posterior distribution
for A, is
 = B(a + 1)
B+1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F059f8a44-0099-41c0-92e3-ad75d014ebad%2F0cf19db6-46b5-429e-8574-997b7be47668%2F0ey7qrj_processed.png&w=3840&q=75)
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