Question 1
MATH40005 Probability and Statistics Resit Preparation
Question 1
1(a). On Ω = {1, 2, 3, 4}, give a nontrivial sigma-algebra F and a probability measure P on it with P({1, 2}) = 0 .4. Verify
the required properties briefly. (4 marks)
1(b). Prove that if events A and B are independent, then Ac and Bc are independent. (3 marks)
1(c). A nine-character password contains exactly three uppercase letters, four digits and two distinct symbols chosen from
seven symbols. Letters and digits may repeat. Count the passwords. (5 marks)
1(d). A deck has 12 cards, of which 5 are winners. Four cards are drawn uniformly without replacement. Find the probability
of drawing exactly two winners. (4 marks)
1(e). Let m ∈ Z and 0 < p < 1. A shifted geometric random variable has proposed pmf
pX (x) = p(1 − p)x−m, x = m, m + 1, . . . .
Verify validity and find its CDF for every real x. (4 marks)
Total: 20 marks
2
Worked solution and marking guidance
MATH40005 Probability and Statistics Resit Preparation
Question 1: solution and marking guide
1(a). Take
F = f∅, Ω, f1, 2g, f3, 4gg,
and define probabilities 0, 1, 0.4, 0.6 respectively. The collection contains ∅, is closed under complements and countable
unions, hence is a sigma-algebra. The assigned probabilities are nonnegative, sum to one on the two atoms, and are countably
additive on disjoint events.
4 marks; A:2, B:2
1(b). Using independence and inclusion–exclusion,
P(Ac \ Bc) = 1 P(A [ B)
= 1 P(A) P(B) + P(A \ B)
= (1 P(A))(1 P(B)) = P(Ac)P(Bc).
3 marks; A:1, B:1, D:1
1(c). Choose the type positions, then fill them:
9!
3!4!2! 263 104 (7 6).
The factor 7 6 assigns two distinct symbols to their ordered positions.
5 marks; A:3, B:1, D:1
1(d). This is hypergeometric:
P(X = 2) =
5
2
7
2
12
4
.
4 marks; A:2, C:1, D:1
1(e). The terms are nonnegative and
∞X
x=m
p(1 p)x−m = p
∞X
k=0
(1 p)k = 1.
For real x,
FX (x) =
(
0, x < m,
1 (1 p)⌊x⌋−m+1, x m.
4 marks; A:2, C:1, D:1
Question total: 20 marks
2
Question 2
MATH40005 Probability and Statistics Resit Preparation
Question 2
2(a). Let
fX,Y (x, y) = c(x2 + y), 0 < y < x < 1,
and zero otherwise. Find c. (3 marks)
2(b). Find the marginal densities of X and Y . (4 marks)
2(c). Find the conditional density of Y given X = x and decide whether X and Y are independent. (3 marks)
2(d). Let X ∼ U(1, 2) and W = X 3. Find the CDF and density of W , and compute E(W ). (6 marks)
2(e). Let X ∼ Poi(λ) and Y ∼ Poi(µ) be independent. Identify and justify the conditional distribution ofX given X+Y = n.
(4 marks)
Total: 20 marks
3
Worked solution and marking guidance
MATH40005 Probability and Statistics Resit Preparation
Question 2: solution and marking guide
2(a). Normalisation gives
1 = c
Z 1
0
Z x
0
(x2 + y) dy dx = c
Z 1
0
x3 + x2
2
dx = c 5
12 ,
so c = 12/5.
3 marks; A:3
2(b). For 0 < x < 1,
fX (x) = 12
5
x3 + x2
2
= 6
5 x2(2x + 1).
For 0 < y < 1,
fY (y) = 12
5
Z 1
y
(x2 + y) dx = 12
5
1 y3
3 + y y2
.
4 marks; A:2, B:2
2(c). For 0 < x < 1 and 0 < y < x ,
fY |X=x(y j x) = x2 + y
x3 + x2/2 .
It depends on x, and the support 0 < y < x also depends on x, so X and Y are not independent.
3 marks; A:1, B:2
2(d). Since x 7! x3 is increasing,
FW (w) =
8
><
>:
0, w < 1,
w1/3 1, 1 w < 8,
1, w 8.
Thus
fW (w) = 1
3w2/3 , 1 < w < 8.
By LOTUS,
E(W ) =
Z 2
1
x3 dx = 24 1
4 = 15
4 .
6 marks; A:1, C:2, D:3
2(e). For x = 0, . . . , n,
P(X = x j X + Y = n) = e−λλx/x! e−µµn−x/(n x)!
e−(λ+µ)(λ + µ)n/n!
=
n
x
λ
λ + µ
x µ
λ + µ
n−x
.
Hence X j (X + Y = n) Bin(n, λ/(λ + µ)).
4 marks; A:1, B:1, C:1, D:1
Question total: 20 marks
3
Question 3
MATH40005 Probability and Statistics Resit Preparation
Question 3
3(a). For a random variable X with finite first and second moments, identify a minimiser of E[(X − a)2], the minimum
value, and a minimiser of E|X − b|. (3 marks)
3(b). Prove the bias–variance decomposition for an estimator bθ of a fixed parameter θ. (3 marks)
3(c). A normal population has known standard deviation 5. For n = 25 , ¯x = 102 . (i) Compute a 95% confidence interval
for µ. (ii) Test H0 : µ = 100 against H1 : µ ̸= 100, giving the p-value and conclusion at level 0.05. (7 marks)
3(d). Define a p-value and explain its relationship to a pre-specified significance threshold α and T ype I error. (4 marks)
3(e). Using Chebyshev’s inequality, find a sample size sufficient to estimate any Bernoulli proportion within 0.05 with
confidence at least 0.96. (3 marks)
Total: 20 marks
4
Worked solution and marking guidance
MATH40005 Probability and Statistics Resit Preparation
Question 3: solution and marking guide
3(a). The squared-loss minimiser is a = EX, and the minimum is Var (X). Any median of X minimises absolute loss
EjX bj.
3 marks; A:2, B:1
3(b). Write
bθ θ =
bθ Ebθ
+
Ebθ θ
.
Squaring and taking expectations eliminates the cross term, giving
E(bθ θ)2 = Var(bθ) + Bias(bθ)2.
3 marks; A:1, B:1, D:1
3(c). The standard error is 5/
p
25 = 1 . The interval is
102 1.96 = (100 .04, 103.96).
For the test,
z = 102 100
1 = 2.00.
The two-sided p-value is 2f1 Φ(2)g 0.0455, so reject H0 at level 0.05.
7 marks; A:3, B:1, C:1, D:2
3(d). A p-value is the smallest significance level at which the observed test result would be rejected, equivalently the null
probability of a result at least as extreme as observed. Reject when p α. For a valid level-α test, the probability of rejection
when H0 is true is at most α; for an exact continuous test it equals α.
4 marks; A:1, B:2, C:1
3(e). For the sample proportion, Var (bp) = p(1 p)/n 1/(4n). Hence
P(jbp pj 0.05) 1/(4n)
0.052 = 100
n .
To make this at most 0.04, take n 2500.
3 marks; A:1, B:1, D:1
Question total: 20 marks
4
Question 4
MATH40005 Probability and Statistics Resit Preparation
Question 4
4(a). Independent observations are uniformly distributed on [θ, θ + 3]. Find the full set of maximum likelihood estimates of
θ in terms of xmin and xmax. (5 marks)
4(b). Let Xi | θ be independent Poi (θ) observations and let θ ∼ Gamma(a, b), where b is a rate. Derive the posterior
distribution. (5 marks)
4(c). In simple linear regression with fixed xi, derive Cov( bβ0, bβ1). (5 marks)
4(d). A fitted linear model has the residual plot below. State whether it fits well and give an appropriate next step. (3 marks)
4(e). The observed sample has size n. State precisely how a nonparametric bootstrap sample is generated and why repeated
values are legitimate. (2 marks)
0 5 10 15 20
−1
−0.5
0
0.5
1
xi
Residual bεi
Total: 20 marks
5
Worked solution and marking guidance
MATH40005 Probability and Statistics Resit Preparation
Question 4: solution and marking guide
4(a). The likelihood equals 3−n when every observation lies in [θ, θ + 3], and zero otherwise. Feasibility requires
θ xmin, θ xmax 3.
Under the stated model, xmax xmin 3 almost surely, and the full set of MLEs is
bθ 2 [xmax 3, xmin].
If an externally supplied dataset had range greater than 3, it would be incompatible with this model.
5 marks; A:2, C:2, D:1
4(b). The likelihood kernel is
L(θ) / θ
∑
i xi e−nθ.
Multiplying by the prior kernel θa−1e−bθ gives
π(θ j x) / θa+∑
i xi−1e−(b+n)θ.
Hence
θ j x Gamma
a +
X
i
xi, b + n
!
in shape–rate parameterisation.
5 marks; A:1, B:1, C:2, D:1
4(c). Usingbβ0 = ¯Y ¯xbβ1, Cov( ¯Y ,bβ1) = 0 , and Var (bβ1) = σ2/Sxx,
Cov(bβ0,bβ1) = ¯x σ2
Sxx
.
The zero covariance follows fromP
i(xi ¯x) = 0 and independent equal-variance errors.
5 marks; A:1, B:1, C:1, D:2
4(d). The model does not fit well because the residual variance grows markedly with x. The constant-variance assumption
fails. Consider a response transformation, weighted least squares, or an explicit variance model, and then inspect the new
residuals.
3 marks; B:3
4(e). Draw n observations independently with replacement from the empirical distribution that assigns probability 1/n to
each observed data point. Repetitions are legitimate because sampling is with replacement; they represent repeated draws
from the empirical distribution.
2 marks; A:2
Question total: 20 marks
5