If X has a Poisson distribution and the prior dis-tribution of its parameter (capital Greek lambda) is a gamma distribution with the parameters α and β,show that(a) the posterior distribution of given X = x is agamma distribution with the parameters α + x and ββ + 1;(b) the mean of the posterior distribution of is μ1 = β(α + x)β + 1

Suppose the random variable, X, follows a geometric distribution with parameter (0< 0 <1). Let X₁, X₂, X, be a random sample of size n from the population of X. (d) Find the method of moments estimator (mme) of 0. (e) Show how the mle of 0 is related to its mme. Explain if it is always true. (f) Verify if X, is a sufficient statistic for 8 or not. (g) Justify if = is an unbiased estimator for 6.

Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)

For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.