Level 2 — RecallProbability & Statistics

Probability & Statistics

⏱ 30 minutes40 marksprintable — key stays hidden on paper

Chapter: 1.3 Probability & Statistics Difficulty: Level 2 (Recall / standard textbook problems) Time limit: 30 minutes Total marks: 40

Answer all questions. Show working where required. Use $...$ notation for math.

Q1. State the three Kolmogorov axioms of probability for a sample space $\Omega$ with probability measure $P$ . (3 marks)

Q2. A fair six-sided die is rolled once. Let $A$ = "outcome is even" and $B$ = "outcome is greater than 3". Compute $P(A)$ , $P(B)$ , and $P(A \cap B)$ . Are $A$ and $B$ independent? Justify. (4 marks)

Q3. A test for a disease has sensitivity $P(T^+\mid D)=0.99$ and specificity $P(T^-\mid D^c)=0.95$ . The disease prevalence is $P(D)=0.01$ . Using Bayes' theorem, compute the probability that a person who tests positive actually has the disease, $P(D\mid T^+)$ . (5 marks)

Q4. A discrete random variable $X$ has pmf $P(X=k)=ck$ for $k\in\{1,2,3,4\}$ and $0$ otherwise. (a) Find $c$ . (2 marks) (b) Compute $E[X]$ . (2 marks) (c) Compute $\operatorname{Var}(X)$ . (2 marks)

Q5. A continuous random variable has pdf $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ ( $\lambda>0$ ). (a) Derive its cumulative distribution function $F(x)$ . (2 marks) (b) State $E[X]$ and $\operatorname{Var}(X)$ for this exponential distribution. (2 marks)

Q6. For a Binomial random variable $X\sim \text{Bin}(n,p)$ , state $E[X]$ and $\operatorname{Var}(X)$ . Then, for $n=10$ , $p=0.3$ , compute $P(X=2)$ . (4 marks)

Q7. Define the entropy $H(X)$ of a discrete distribution. Compute the entropy (in bits) of a Bernoulli random variable with $p=0.5$ . (3 marks)

Q8. State the Central Limit Theorem for the sample mean $\bar{X}_n$ of $n$ i.i.d. random variables with mean $\mu$ and finite variance $\sigma^2$ . Include the limiting distribution. (3 marks)

Q9. In a Poisson process, calls arrive at an average rate of $\lambda=3$ per minute. Compute the probability of exactly 2 calls in a given minute. (3 marks)

Q10. A sample of $n=100$ observations has sample mean $\bar{x}=50$ and known population standard deviation $\sigma=10$ . Construct a 95% confidence interval for the population mean $\mu$ (use $z_{0.025}=1.96$ ). (4 marks)

End of paper

Answer keyMark scheme & solutions

Q1. (3 marks)

(i) Non-negativity: $P(A)\ge 0$ for all events $A$ . (1)
(ii) Normalization: $P(\Omega)=1$ . (1)
(iii) Countable additivity: for disjoint events $A_1,A_2,\dots$ , $P\!\left(\bigcup_i A_i\right)=\sum_i P(A_i)$ . (1) Why: These axioms ensure probabilities are consistent, bounded in $[0,1]$ , and additive over mutually exclusive outcomes.

Q2. (4 marks)

$A=\{2,4,6\}\Rightarrow P(A)=3/6=1/2$ . (1)
$B=\{4,5,6\}\Rightarrow P(B)=3/6=1/2$ . (1)
$A\cap B=\{4,6\}\Rightarrow P(A\cap B)=2/6=1/3$ . (1)
Independence check: $P(A)P(B)=1/4 \ne 1/3=P(A\cap B)$ , so not independent. (1)

Q3. (5 marks) By Bayes: $P(D\mid T^+)=\dfrac{P(T^+\mid D)P(D)}{P(T^+\mid D)P(D)+P(T^+\mid D^c)P(D^c)}$ . (2)

$P(T^+\mid D^c)=1-0.95=0.05$ ; $P(D^c)=0.99$ . (1)
Numerator $=0.99\times0.01=0.0099$ . Denominator $=0.0099+0.05\times0.99=0.0099+0.0495=0.0594$ . (1)
$P(D\mid T^+)=0.0099/0.0594\approx 0.1667$ (≈16.7%). (1) Why: Low prevalence dominates, so even an accurate test yields low posterior — the base-rate effect.

Q4. (6 marks) (a) $\sum ck = c(1+2+3+4)=10c=1\Rightarrow c=1/10$ . (2) (b) $E[X]=\sum k\,P(X=k)=\tfrac{1}{10}(1+4+9+16)=30/10=3$ . (2) (c) $E[X^2]=\tfrac{1}{10}(1\cdot1+4\cdot2+9\cdot3+16\cdot4)=\tfrac{1}{10}(1+8+27+64)=100/10=10$ . $\operatorname{Var}=10-3^2=1$ . (2)

Q5. (4 marks) (a) $F(x)=\int_0^x \lambda e^{-\lambda t}dt=1-e^{-\lambda x}$ for $x\ge0$ , $0$ otherwise. (2) (b) $E[X]=1/\lambda$ , $\operatorname{Var}(X)=1/\lambda^2$ . (2)

Q6. (4 marks)

$E[X]=np$ , $\operatorname{Var}(X)=np(1-p)$ . (2)
$P(X=2)=\binom{10}{2}(0.3)^2(0.7)^8=45\times0.09\times0.05764801\approx0.2335$ . (2)

Q7. (3 marks)

$H(X)=-\sum_x P(x)\log_2 P(x)$ . (1)
For $p=0.5$ : $H=-(0.5\log_2 0.5 + 0.5\log_2 0.5)=-(0.5(-1)+0.5(-1))=1$ bit. (2)

Q8. (3 marks) As $n\to\infty$ , $\dfrac{\bar{X}_n-\mu}{\sigma/\sqrt{n}}\xrightarrow{d}\mathcal{N}(0,1)$ . (2) Equivalently $\bar{X}_n\approx\mathcal{N}(\mu,\sigma^2/n)$ for large $n$ , regardless of the underlying distribution (finite variance). (1)

Q9. (3 marks) $P(X=2)=\dfrac{\lambda^2 e^{-\lambda}}{2!}=\dfrac{9e^{-3}}{2}=4.5\,e^{-3}\approx 0.2240$ . (3)

Q10. (4 marks) CI $=\bar{x}\pm z\,\dfrac{\sigma}{\sqrt{n}}=50\pm1.96\cdot\dfrac{10}{\sqrt{100}}=50\pm1.96\cdot1=50\pm1.96$ . (3) Interval: $(48.04,\ 51.96)$ . (1)

[
  {"claim":"Q3 Bayes posterior ≈0.1667","code":"num=0.99*0.01; den=num+0.05*0.99; result=abs(num/den-0.16666666666666666)<1e-6"},
  {"claim":"Q4 c=1/10, E[X]=3, Var=1","code":"c=Rational(1,10); ks=[1,2,3,4]; EX=sum(k*c*k for k in ks); EX2=sum(k*k*c*k for k in ks); result=(sum(c*k for k in ks)==1) and (EX==3) and (EX2-EX**2==1)"},
  {"claim":"Q6 Binomial P(X=2)≈0.2335","code":"from sympy import binomial; p=binomial(10,2)*Rational(3,10)**2*Rational(7,10)**8; result=abs(float(p)-0.233474)<1e-4"},
  {"claim":"Q9 Poisson P(X=2)≈0.2240","code":"val=Rational(9,2)*exp(-3); result=abs(float(val)-0.224042)<1e-4"},
  {"claim":"Q10 CI is (48.04,51.96)","code":"lo=50-1.96*10/ (100**0.5); hi=50+1.96*10/(100**0.5); result=(abs(lo-48.04)<1e-9) and (abs(hi-51.96)<1e-9)"}
]