1. A-F
Suppose we observe x_(1),x_(2),dots,x_(n) as independent and identically distributed as an exponen-
tial random variable with f_(lambda )(x_(1))=lambda e^(lambda _(1)), and recall that E(x_(1))=(1)/(lambda ) and Var(x_(1))=(1)/(lambda ^(2)).
(a) Write down the conditional density f(x_(1)=x_(1),x_(2)=x_(2),dots,x_(n)=x_(n)|lambda ); essentially the
likelihood function.
(b) Suppose we assign a prior distribution to lambda based on the gamma distribution. In partic-
ular, let lambda ∼Gamma (alpha ,eta ) so that the probability density function of lambda is
f_(alpha _(eta )eta )(lambda )=(eta ^(alpha ))/(Gamma (alpha ))lambda ^(alpha 1_(e))eta lambda .
Recall that E(lambda )=(alpha )/(eta ) and Var(lambda )=(alpha )/(eta ^(2)) for this gamma distribution. Find the posterior
distribution of lambda |x_(1),x_(2),dots,x_(n). One may use prop to avoid computing f(x_(1),dots,x_(n)).
(c) Based on the distribution found in part (b), compute what is called the posterior mean
of lambda :
E(lambda |x_(1),x_(2),dots,x_(n)).
What is the effect of larger versus smaller values of sum_(i-1)^n x_(i) on the posterior mean?
Does this make sense?
(d) When n grows, we might expect that sum_(i=1)^n x_(i) grows. If n and sum_(i-1)^n x_(i) are very large
and becoming larger, what does the posterior mean converge to?
(e) Based on the distribution found in part (b), compute what is called the posterior variance
of lambda :
Var(lambda |x_(1),x_(2),dots,x_(n)).
(f) If n and sum_(i-1)^n x_(i) are very large and becoming larger, what does the posterior variance
converge to?
1. Suppose we observe X1, X2, ..., X, as independent and identically distributed as an exponen-
(c) Based on the distribution found in part (b), compute what is called the posterior mean of A:
(a) Write down the conditional density f(, = ,X = 2,..,X, = ,|); essentially the likelihood function.
E(|X,X2,...,X,).
What is the effect of larger versus smaller values of o-, X, on the posterior mean? Does this make sense?
(d) When n grows, we might expect that , X; grows. If n and , X; are very large and becoming larger, what does the posterior mean converge to?
(b) Suppose we assign a prior distribution to A based on the gamma distribution. In partic ular, let Gamma(, ) so that the probability density function of is
1e
Recall that E() = and Var() = for this gamma distribution. Find the posterior distribution of |X,X,...,,. One may use to avoid computing f(,...,X,).
(e) Based on the distribution found in part (b), compute what is called the posterior variance of A:
Var(|X,X2,...,X,.
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e
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(f) If n and , X, are very large and becoming larger, what does the posterior variance converge to?