Probability Theory & Statistics

Level: 1 — Recognition (MCQ + Matching + True/False with justification) Time limit: 20 minutes Total marks: 30

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Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. Which is NOT one of Kolmogorov's axioms of probability? (a) $P(\Omega)=1$ (b) $P(A)\ge 0$ (c) Countable additivity for disjoint events (d) $P(A\cup B)=P(A)+P(B)$ for all events

Q2. For events $A,B$, the inclusion–exclusion principle gives $P(A\cup B)=$ (a) $P(A)+P(B)$ (b) $P(A)+P(B)-P(A\cap B)$ (c) $P(A)P(B)$ (d) $P(A)+P(B)+P(A\cap B)$

Q3. If $X\sim\text{Binomial}(n,p)$, then $E[X]$ and $\text{Var}(X)$ are (a) $np,\ np$ (b) $np,\ np(1-p)$ (c) $p,\ p(1-p)$ (d) $np(1-p),\ np$

Q4. The MGF of a random variable $X$ is defined as (a) $E[X^t]$ (b) $E[e^{tX}]$ (c) $E[t^X]$ (d) $\ln E[e^{tX}]$

Q5. For $X\sim\text{Poisson}(\lambda)$, $E[X]$ equals (a) $\lambda^2$ (b) $\sqrt{\lambda}$ (c) $\lambda$ (d) $1/\lambda$

Q6. The PDF of $X\sim\text{Exponential}(\lambda)$ (rate form), for $x\ge 0$, is (a) $\lambda e^{-\lambda x}$ (b) $e^{-x/\lambda}$ (c) $\frac{1}{\lambda}e^{-\lambda x}$ (d) $\lambda x e^{-\lambda x}$

Q7. A $t$-distribution with $k$ degrees of freedom arises as (a) sum of $k$ squared normals (b) $Z/\sqrt{\chi^2_k/k}$ with $Z,\chi^2_k$ independent (c) ratio of two chi-squares (d) product of $k$ normals

Q8. Two random variables $X,Y$ are independent if and only if (a) $\text{Cov}(X,Y)=0$ (b) $E[XY]=E[X]E[Y]$ (c) $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ for all $x,y$ (d) they have equal variance

Q9. The correlation coefficient $\rho$ always satisfies (a) $0\le\rho\le1$ (b) $-1\le\rho\le1$ (c) $\rho\ge0$ (d) $\rho$ unbounded

Q10. The Central Limit Theorem states that for i.i.d. $X_i$ with mean $\mu$, variance $\sigma^2$, the quantity $\frac{\bar X_n-\mu}{\sigma/\sqrt n}$ converges in distribution to (a) $\text{Exponential}(1)$ (b) $N(0,1)$ (c) $\chi^2_1$ (d) $\text{Uniform}(0,1)$

Q11. An estimator $\hat\theta$ is unbiased if (a) $\text{Var}(\hat\theta)\to0$ (b) $E[\hat\theta]=\theta$ (c) $\hat\theta\to\theta$ a.s. (d) $\hat\theta$ is the MLE

Q12. In Bayesian statistics, posterior $\propto$ (a) prior $\times$ likelihood (b) prior $+$ likelihood (c) likelihood only (d) prior only

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Section B — Matching (1 mark each)

Q13–Q17. Match each distribution/quantity (left) to its correct property (right). Write pairs like 13→(x).

#	Item
13	$\text{Var}(X)$ for $X\sim\text{Geometric}(p)$ (support $1,2,\dots$)
14	Mean of $\text{Uniform}(a,b)$
15	MGF of $N(\mu,\sigma^2)$
16	Degrees of freedom of chi-squared sum of $n$ standard normals squared
17	Method of moments principle

Option	Property
(p)	$\frac{a+b}{2}$
(q)	$\exp(\mu t+\tfrac12\sigma^2 t^2)$
(r)	$\frac{1-p}{p^2}$
(s)	$n$
(t)	equate sample moments to population moments

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Section C — True/False WITH justification (2 marks each: 1 mark T/F, 1 mark justification)

Q18. "If $\text{Cov}(X,Y)=0$ then $X$ and $Y$ are independent." True or False? Justify.

Q19. "For any random variable, $\text{Var}(aX+b)=a^2\text{Var}(X)$." True or False? Justify.

Q20. "A 95% confidence interval means the true parameter lies in the computed interval with probability 0.95." True or False? Justify.

Q21. "The p-value is the probability that the null hypothesis is true." True or False? Justify.

Q22. "By the Weak Law of Large Numbers, the sample mean converges in probability to the population mean." True or False? Justify.

[
{"claim":"Binomial(n,p) variance = np(1-p), check n=10,p=0.3 gives 2.1","code":"n,p=10,Rational(3,10); result = (n*p*(1-p))==Rational(21,10)"},
{"claim":"Uniform(a,b) mean = (a+b)/2 for a=2,b=8 is 5","code":"a,b=2,8; result = Rational(a+b,2)==5"},
{"claim":"Geometric(p) variance (1-p)/p^2 for p=1/4 equals 12","code":"p=Rational(1,4); result = ((1-p)/p**2)==12"},
{"claim":"Normal MGF exp(mu t + sigma^2 t^2/2) has second derivative at 0 = mu^2+sigma^2","code":"t,mu,s=symbols('t mu s'); M=exp(mu*t+s**2*t**2/2); result = simplify(diff(M,t,2).subs(t,0)-(mu**2+s**2))==0"}
]