程序代写案例-STAT0005
时间:2022-04-09
Examination Paper for STAT0005 Page 3
Section A
• A1 Let X ⇠ U(1, 1) and let Y = X4.
(a) Compute E[Y ]. [3]
(b) Compute Var(Y ). [3]
(c) Compute the pdf fY of Y . [5]
• A2 Let the joint distribution of X and Y be given by the following
two-way table:
X
Y -1 0 1
-1 b 0 a
0 0 1-2a-2b 0
1 a 0 b
Here, a, b are unknown constants such that the above table is a valid
two-way table. For parts (a)-(d) you may leave your results in terms
of a and b where necessary.
(a) Compute the marginal pmfs pX and pY . [3]
(b) Compute E[X] and E[Y ]. [3]
(c) Compute Var(X) and Var(Y ). [3]
(d) Compute Corr(X, Y ) in the case (a, b) 6= (0, 0). [5]
(e) What constraints must the pair (a, b) satisfy to ensure that the
table above is valid? [2]
(f) What is the smallest value Corr(X, Y ) can take in this case? Give
a value of (a, b) for which the smallest possible value of Corr(X, Y )
is attained. [2]
• A3 Let the joint cdf of X and Y be given by
FX,Y (x, y) =
8<: 0 if x < 0 or y < 0min{x, y} if x, y 0 and (x  1 or y  1)
1 if x, y 1
.
(a) Compute P (0 < X  1, 0 < Y  1). [2]
(b) Compute the marginal cdf FY of Y . [3]
(c) Compute P (X < 1/2 |Y < 1/2).
[Type] Using only words and no formulae, decide whether X and
Y are independent and justify your decision. Maximum length:
150 words. [5]
Turn Over
Examination Paper for STAT0005 Page 4
Section B
• B1 For i 2 {1, . . . , n} with n 2 N and n 2, let Xi i.i.d.⇠ N(µ, 2) where
µ and 2 > 0 are both unknown. To estimate the variance 2, consider
the estimator
T↵ =
1
n ↵
nX
i=1
(Xi X¯)2,
where X¯ = 1n
Pn
i=1Xi. In this expression, the number ↵ 2 (1, 2) is
used to obtain di↵erent estimators.
(a) Name the estimator in the case ↵ = 1. What is the expected
value of T1? (You do not need to compute the expected value if
you know it.) [2]
(b) Compute the bias of T↵. [2]
(c) Compute the sampling variance of T↵. Hint: Start from the
variance of T1. [5]
(d) Show that the mse for T↵ is given by
mse(T↵;
2) =
4
(n ↵)2

↵2 2↵ 1 + 2n .
[4]
(e) Given the sample size n, which value ↵⇤(n) of ↵ results in the
smallest mean square error of T↵? You need to provide a derivation
and justification of your result and while you may omit checking
the second order condition you should check the boundaries. [8]
(f) Hence provide a formula for an estimator of the form T↵ with
smallest mse.
[Type] Using only words and no formulae, give one reason why T1
is often used in practice in spite of your result. Maximum length:
300 words. [5]
Continued
Examination Paper for STAT0005 Page 5
• B2 For n 2 N, consider the regression model of the form
Y = ↵x+ ✏, ✏ ⇠ N(0,⌃),
where the covariate x 2 Rn with x 6= 0 and the positive definite sym-
metric matrix ⌃ 2 Rn⇥n are fixed and known whereas the parameter
↵ 2 R is unknown. The sample consists of a single observation y from
this model.
(a) [Type] Using only words and no formulae, explain how a maxi-
mum likelihood estimator is obtained in general. Maximum length:
150 words. [4]
(b) In the setting described above, obtain the log-likelihood for the
parameter ↵ given the one observation y 2 Rn. [4]
(c) Show that the MLE of ↵ is given by
b↵MLE = xT⌃1y
xT⌃1x
.
You need to check all applicable conditions for a maximum.
Hint: Note that ↵ is a number (not a matrix or a vector) and
note the dimension of xT⌃1y as well as that of xT⌃1x. [8]
(d) Compute E[b↵MLE]. [5]
(e) Compute the Cramer-Rao lower bound for unbiased estimators of
↵. What is the interpretation of this bound? [4]
(f) Compute the sampling variance of b↵MLE. Hence decide whether
or not b↵MLE achieves the Cramer-Rao lower bound.
Hint: First show that b↵MLE is of the form b↵MLE = c+ aT ✏ for
some constant c 2 R and some constant vector a 2 Rn which you
should specify.
[8]
Turn Over
Examination Paper for STAT0005 Page 6
Section C
Let X ⇠ N(µ,⌃) where µ 2 Rn and ⌃ 2 Rn⇥n for dimension n 2.
Also assume that ⌃ is positive definite symmetric. Let A 2 Rk⇥n with
k 2 {1, . . . , n}. You may use the facts that, firstly, A⌃AT is positive definite
symmetric if A has full rank and that, secondly, A has full rank if and only
if its row vectors are linearly independent.
(a) Compute the mgf of Y = AX and thus show that Y follows a normal
distribution and specify its mean vector and covariance matrix. [4]
(b) Under the condition that A has full rank, write down the pdf of Y .
Explain why it is impossible to write down the pdf of Y if A does not
have full rank. [4]
(c) Using the mgf or otherwise, prove that Cov(Xi, Xj) = 0 if and only if
Xi and Xj are independent. Note that during the course we have only
established this in the case of bivariate normal distributions. [6]
(d) Suppose that Cov(X1, Xj) = 0 holds for all j 2 {2, . . . , d}. Show that
X1 is independent of X2 +X3 + . . .+Xd. [5]
(e) Consider three discrete random variables U, V,W each taking values in
{1, 1} and such that U is independent of V and U is independent of
W . For each of the statements (i) and (ii) below, decide whether it
is true in general or whether it may be false. If the statement is true
in general, provide a proof. Otherwise, find a joint pmf for U, V and
W that provides a counterexample (i.e. a joint distribution for which
the statement does not hold), explaining your reasoning and presenting
your joint pmf in a table as follows:
u v w pUVW (u, v, w)
-1 -1 -1 . . .
1 -1 -1 . . .
...
...
...
(i) U and V +W are uncorrelated. [4]
(ii) U and V +W are independent. [5]
(f) [Type] Using only words and no formulae, write a short essay on three
ways in which the Gaussian distribution is special among probability
distributions. You need to make clear how the ways are directly related
to this Section C and/or to results in the lecture notes. Maximum
length: 300 words. [10]
End of Paper


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