STAT 522
Practice Midterm Exam
1. Let Xl, X 2, " ' J X n be n independent samples from N(f.l, 0- 2). Let 11, l2, ... , In and ml, m2, ... , m n
2
,\,n
2
Defi ne
. f'
be k nown constants satls
ymg ,\,n
LJi=1 li = ,\,n
LJi=1 mi = 0, ,\,n
LJi=1 li = LJi=l m i = 1.
U = 2::r=1 liXi and V = 2::r=1 miXi·
(a) State the joint distribution of U and V. Under what conditions are these two random
variables independent?
(b) Under the condition obtained in (a), state the distribution of the following statistics:
U2
V2 '
U2
U 2 + V2
U
I
IVT '
2. (a) Suppose X is a Binomial(n,p) random variable, 0 < P < 1. Find the m.l.e. for p(l-p).
Show that the m.l.e. is not unbiased for p(l - p). Construct an unbiased estimator for
p(l - p) using this m .1.e.
(b) Suppose X is a Binomial(2,p) random variable where p can take only two values! and
Show that this family is not complete by constructing a non-zero function g(X) whose
expectation is zero for both p = ~ and
t.
t.
3. Let Xl, X2, X 3 be a random sample of size three from the uniform distribution U(O, B), where
e > 0 is an unknown parameter. Let X(l), X(2), X(3) be the corresponding order statistics.
(a) Find the marginal pdf X(l) and show that
x(1)/e is distributed as Beta(l, 3).
(b) Compute E[X(1)]. Construct an unbiased estimator for
e usiug X(l)'
(c) Show that X(3)/ X(l) is independent of X(3).
4. Let X l ,X2 ,,,,,Xn be a random sample from Unif(aB,b{)), where a < b are positive constants
and B > 0 is an unknown parameter.
(a) Find a minimal sufficient statistic for B.
(b) Is the minimal statistic found in part (a) complete? Justify your answer.
(c) Find the m.l.e for
e.
(d) Find the m.l.e for population median.
5. Let X I ,X2,,,,,Xn be a random sample from a distribution with an exponential distribution
f(xIB) = Be-ex,
where
x> 0, e > 0
Suppose the prior distribution of B is Gamma( a,,O).
(a) Find the posterior density function of
e given Xl, ... , X n .
e
(b) The Bayes estimator of for the squared error loss function is the posterior mean of B.
Compute the Bayes estimator of B.
(c) Explain why the posterior distribution of B given Xl, ... , X n is the same as the posterior
distribution of B given 2::r=1 Xi.
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1. Let X1 , ..., Xn be iid Poisson (λ).
(a) Find the UMVUE for λ.
(b) Find the MLE for e−λ .
(c) Find the UMVUE for e−λ .
2. Let X1 , ..., Xn be iid Normal (µ, 1), with µ unknown. Let −∞ < c < ∞
be a constant.
(a) Find the MLE for P (X1 ≤ c).
(b) Show that (X1 − X̄) ∼ N (0, n−1
). Note that X1 is not independent
n
of X̄.
(c) Show that (X1 − X̄) is independent of X̄.
(d) Find the UMVUE for P (X1 ≤ c).
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