Level 2 — RecallProbability Theory & Statistics

Probability Theory & Statistics

40 marksprintable — key stays hidden on paper

Level: 2 (Recall: definitions, standard problems, short derivations) Time: 30 minutes Total marks: 40

Q1. State the three Kolmogorov axioms of a probability measure $P$ on a sample space $\Omega$ with sigma-algebra $\mathcal{F}$ . (3 marks)

Q2. A fair six-sided die is rolled once. Let $A=\{\text{even}\}$ and $B=\{\text{greater than }3\}$ . Use the inclusion–exclusion principle to compute $P(A\cup B)$ . (4 marks)

Q3. A discrete random variable $X$ has PMF $p(x)=cx$ for $x=1,2,3,4$ and $0$ otherwise. (a) Find $c$ . (b) Compute $E[X]$ . (4 marks)

Q4. Let $X\sim\text{Binomial}(n=10,\,p=0.3)$ . State $E[X]$ and $\mathrm{Var}(X)$ , and compute $P(X=2)$ (leave in exact form). (5 marks)

Q5. Let $X\sim\text{Exponential}(\lambda)$ with PDF $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ . (a) Derive the CDF. (b) State the median in terms of $\lambda$ . (4 marks)

Q6. Define the moment generating function $M_X(t)$ of a random variable $X$ . For $X\sim\text{Poisson}(\mu)$ , it is known that $M_X(t)=e^{\mu(e^t-1)}$ . Use it to obtain $E[X]$ . (4 marks)

Q7. For random variables $X,Y$ , define the covariance $\mathrm{Cov}(X,Y)$ and the correlation coefficient $\rho_{X,Y}$ . State the value of $\rho$ when $X$ and $Y$ are independent. (4 marks)

Q8. State the Central Limit Theorem for a sequence of i.i.d. random variables with mean $\mu$ and finite variance $\sigma^2$ . (4 marks)

Q9. A sample of $n=100$ has mean $\bar{x}=50$ from a population with known $\sigma=10$ . Construct a 95% confidence interval for the population mean (use $z_{0.025}=1.96$ ). (4 marks)

Q10. Define a Type I error and a Type II error in hypothesis testing, and state which one the significance level $\alpha$ controls. (4 marks)

End of paper.

Answer keyMark scheme & solutions

Q1. (3 marks)

(i) Non-negativity: $P(A)\ge 0$ for all $A\in\mathcal{F}$ . (1)
(ii) Normalization: $P(\Omega)=1$ . (1)
(iii) Countable additivity: for pairwise disjoint $A_1,A_2,\dots\in\mathcal{F}$ , $P\!\left(\bigcup_i A_i\right)=\sum_i P(A_i)$ . (1) Why: these axioms guarantee $P$ is a well-defined normalized measure.

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)
$P(A\cup B)=1/2+1/2-1/3=2/3$ . (1)

Q3. (4 marks)

Normalization: $\sum_{x=1}^4 cx=c(1+2+3+4)=10c=1\Rightarrow c=1/10$ . (2)
$E[X]=\sum x\,p(x)=\tfrac{1}{10}(1+4+9+16)=\tfrac{30}{10}=3$ . (2)

Q4. (5 marks)

$E[X]=np=10(0.3)=3$ . (1)
$\mathrm{Var}(X)=np(1-p)=10(0.3)(0.7)=2.1$ . (1)
$P(X=2)=\binom{10}{2}(0.3)^2(0.7)^8$ . (2)
$=45(0.09)(0.7^8)\approx 0.2335$ . (1)

Q5. (4 marks)

CDF: $F(x)=\int_0^x\lambda e^{-\lambda t}\,dt=1-e^{-\lambda x}$ for $x\ge0$ (0 otherwise). (2)
Median $m$ : $F(m)=1/2\Rightarrow 1-e^{-\lambda m}=1/2\Rightarrow m=\dfrac{\ln 2}{\lambda}$ . (2)

Q6. (4 marks)

Definition: $M_X(t)=E[e^{tX}]$ (when it exists in a neighborhood of $0$ ). (1)
$E[X]=M_X'(0)$ . (1)
$M_X'(t)=e^{\mu(e^t-1)}\cdot\mu e^t$ . (1)
At $t=0$ : $M_X'(0)=e^0\cdot\mu\cdot1=\mu$ , so $E[X]=\mu$ . (1)

Q7. (4 marks)

$\mathrm{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]$ . (2)
$\rho_{X,Y}=\dfrac{\mathrm{Cov}(X,Y)}{\sigma_X\sigma_Y}$ . (1)
If $X,Y$ independent then $\mathrm{Cov}=0\Rightarrow\rho=0$ . (1)

Q8. (4 marks)

Let $X_1,\dots,X_n$ i.i.d. with mean $\mu$ , variance $\sigma^2<\infty$ , $\bar{X}_n=\tfrac1n\sum X_i$ . (1)
Then $\dfrac{\bar{X}_n-\mu}{\sigma/\sqrt{n}}\xrightarrow{d} N(0,1)$ as $n\to\infty$ . (2)
Equivalently $\sqrt{n}(\bar{X}_n-\mu)\xrightarrow{d}N(0,\sigma^2)$ . (1)

Q9. (4 marks)

CI: $\bar{x}\pm z_{0.025}\dfrac{\sigma}{\sqrt{n}}$ . (1)
Margin $=1.96\cdot\dfrac{10}{\sqrt{100}}=1.96\cdot1=1.96$ . (2)
Interval: $(48.04,\,51.96)$ . (1)

Q10. (4 marks)

Type I error: rejecting $H_0$ when $H_0$ is true (false positive). (1.5)
Type II error: failing to reject $H_0$ when $H_0$ is false (false negative). (1.5)
$\alpha$ (significance level) controls the Type I error probability. (1)

[
  {"claim":"Q2: P(A∪B)=2/3","code":"PA=Rational(1,2); PB=Rational(1,2); PAB=Rational(1,3); result = (PA+PB-PAB == Rational(2,3))"},
  {"claim":"Q3: c=1/10 and E[X]=3","code":"c=Rational(1,10); E=sum(x*c*x for x in [1,2,3,4]); result = (10*c==1 and E==3)"},
  {"claim":"Q4: P(X=2) ≈ 0.2335","code":"from sympy import binomial, Rational; p=binomial(10,2)*Rational(3,10)**2*Rational(7,10)**8; result = (abs(float(p)-0.2335)<0.001)"},
  {"claim":"Q9: 95% CI is (48.04,51.96)","code":"m=1.96*10/ (100**0.5); result = (abs((50-m)-48.04)<1e-9 and abs((50+m)-51.96)<1e-9)"}
]