Examination Paper for STAT0005, Level 5
Page 1
STAT0005: Probability and Inference
Level 5, 2018/19
Answer ALL questions. Section A carries 40% of the total marks and Section
B carries 60%. The relative weights attached to each question are as follows:
A1(10), A2(11), A3(9), A4(10), B1(31), B2(29). The numbers in square
brackets indicate the relative weight attached to each part question.
You may use without proof the following results:
ˆ A random variable X follows the Poisson distribution with parameter
µ > 0, denoted X ∼ Poi(µ), if and only if it has pmf
P (X = k) = e−µ
µk
,
k!
k ∈ {0, 1, 2, 3, . . .}.
In this case E[X] = µ, Var(X) = µ and GX (z) = exp (µ(z − 1)).
ˆ A random variable X follows the Geometric distribution with parameter p ∈ (0, 1), denoted X ∼ Geo(p), if and only if it has pmf
P (X = k) = (1 − p)k−1 p,
In this case, E[X] =
k ∈ N.
1
1−p
and Var(X) =
.
p
p2
ˆ A random variable X follows the Binomial distribution with parameters
n ∈ N and p ∈ (0, 1), denoted X ∼ Bin(n, p), if and only if it has pmf
n k
P (X = k) =
p (1 − p)n−k , k ∈ {0, 1, 2, . . . , n}.
k
In this case E[X] = np and Var(X) = np(1 − p).
ˆ A random variable X follows the Gamma distribution with parameters
α > 0 and λ > 0 if it has pdf
 α α−1 −λx
 λ x e
if x > 0
fX (x) =
.
Γ(α)

0
otherwise
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Examination Paper for STAT0005, Level 5
Page 2
α
α
s −α
, Var(X) = 2 and MX (s) = 1 −
for
λ
λ
R ∞ s−1 λ−x
s < λ. The Gamma function is given by Γ(s) = 0 x e dx; if
s ∈ N then we have Γ(s) = (s − 1)!. Finally, if X ∼ Gam(α, λ) and
Y ∼ Gam(β, λ) are two independent random variables (with α, β, λ >
0) then X + Y ∼ Gam(α + β, λ) follows.
In this case E[X] =
Continued
Examination Paper for STAT0005, Level 5
Page 3
Section A
A1 Let X follow the distribution given by the following pmf:
k
P (X = k)
0
1/4
1
1/4
2
1/4
3
1/4
(a) Compute the cdf and sketch it bearing in mind that it is defined
on the whole real axis.
[6]
(b) Consider the transformed random variable Y = |X − 2|. Compute
its pmf, name the distribution and provide any parameter value(s)
required as well as the expected value and variance.
[4]
A2 Let X ∼ Geo(p) with p ∈ (0, 1). Conditionally on X, Y takes the value
X with probability 1/2 and the value 3X with probability 1/2.
(a) Compute EY |X [Y |X].
[2]
(b) Compute E[Y ].
[3]
(c) Compute VarY |X (Y |X).
[2]
(d) Show that Var(Y ) = 6p−2 − 5p−1 .
[4]
A3 Let X have pmf pX (k) =
µk
for k ∈ N, where µ > 0 is a
k!(eµ − 1)
parameter.
eµz − 1
.
eµ − 1
(b) Hence or otherwise compute E[X].
(a) Show that GX (z) =
[5]
[4]
A4 Let X1 and X2 denote two tosses of a potentially unfair coin, such that
P (X1 = 1) = P (X2 = 1) = p and P (X1 = 0) = P (X2 = 0) = 1 − p for
some p ∈ (0, 1). In this question, all coin tosses are independent.
(a) Let W = (X1 − X2 )2 . Compute the pmf of W .
[3]
(b) Compute P (X1 = 1|W = 1) and P (X1 = 0|W = 1).
[4]
(c) Let Y denote the number of replications of the above two-toss
experiment until W = 1 happens. Note that Y takes values in N.
What is its distribution including any parameter value(s) needed?
[3]
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Examination Paper for STAT0005, Level 5
Page 4
Section B
B1 For n ∈ N, let X1 , X2 , . . . , Xn be i.i.d. random variables each with pdf
f (x) =
2x −x2 /θ
e
θ
for x ∈ (0, ∞),
where θ ∈ (0, ∞) is an unknown parameter.
P
(a) Show that θb = n1 ni=1 Xi2 is the MLE of θ.
[7]
(b) Show that the sampling distribution of θb is Gam(n, nθ−1 ). Hint:
Compute the distribution of X12 first. You may also use the additive property of Gamma-distributed random variables given in
the preamble.
[6]
b
(c) Hence or otherwise compute the bias and variance of θ.
[2]
(d) Compute the CRLB for θ. Does θb achieve the same mse as an
unbiased estimator attaining the CRLB? Justify.
[5]
(e) Obtain an approximate 95% confidence interval for θ based on
b
asymptotic normality of θ.
[3]
b n), b(θ,
b n)] is a 95% confidence
(f) Remembering that an interval [a(θ,
b n), b(θ,
b n)] 3 θ) ≥ 0.95, decide
interval if and only if P ([a(θ,
whether the interval computed in (e) actually is a 95% confidence
interval in the case n = 2. Discuss your findings commenting on
the statistical implications of using the approximate confidence interval in (e) assessing their severity. Hint: Using the sampling
distribution ofθb established in (b),
you will need to calculate the
θ
√ ≤ θb . You may require integration
probability P
1 + 1.96/ 2
by parts and a pocket calculator.
[8]
Continued
Examination Paper for STAT0005, Level 5
Page 5
B2 (a) State the definition of the mgf MX of a random variable X.
[3]
(b) State how the pgf GX and the mgf MX of a discrete random
variable X taking values in N are related.
[3]
(c) What value does MX (s) take at s = 0? Does this depend on the
distribution of X? Justify your answer.
[4]
(d) Explain carefully how mgfs are used to establish independence and
how they are used to decide whether two random variables follow
the same distribution. Your explanation should comment on why
it is important to include the range of s for which MX (s) exists
when reporting the mgf MX .
[7]
(e) Fix k ∈ N and λ > 0. For this question you may use the following
fact without proof:
√
P
i.i.d.
For i ∈ N, let Xi ∼ N ( λ, 1). Then, Y = ki=1 Xi2 has mgf
λks
−k/2
MY (s) = (1 − 2s)
exp
, s ∈ (−∞, 1/2).
1 − 2s
i.i.d.
(i) For i ∈ N, let Zi ∼ χ21 . For j ∈ N, compute the mgf of
P
U = k+2j
i=1 Zi . Name the distribution of U and provide any
parameter value(s) needed.
[6]
(ii) Now let J ∼ Poi(kλ/2) be independent of the Zi . Compute
P
the mgf of W = k+2J
i=1 Zi , decide whether W and Y follow
the same distribution and justify your decision. Hint: Conditioning may help you re-use previous results.
[6]
End of Paper