please do my assignment

Solution 1:
Let be a random sample from exponential distribution with pdf given as follows:
a)
The expected value is calculated as follows:
Now, set the expected value equal to the sample mean that is .
b)
A statistic T is sufficient statistic for , iff the pdf can be partitioned as follows:
Sufficient statistic using factorization theorem,
Thus, by factorization theorem and
Hence, is sufficient statistic.
c)
Let T be the random variable for wait time until the event arrived and the event of arrival is modeled using the Poisson distribution with parameter n and k the parameter of gamma that is the rate of events happening which follows Poisson process.
Thus, be the probability that the waiting time until the event is greater than the t time units.
Thus, first we find the CDF and then...