MA2510-2425A Solution

Certainly! Below, each question from your attached MA2510 Probability & Statistics exam is restated, then analyzed with a layered, hierarchical breakdown. Each layer summarizes the underlying theory, formulas, and reasoning, mapping the question content directly to the logical/statistical concepts required to solve it.

Q1. Decks of Cards & Probability (Combinatorics, Conditional Probability, Discrete Distributions)

Restatement

1. Two decks: a black deck (52 cards), red deck (32 cards).
   • (i) Toss coin, pick deck; then pick 2 cards simultaneously. Find probability two cards are of the same value.
   • (ii) Given two cards of the same value, probability black deck was chosen?
   • (iii) Merge & shuffle both decks (84 cards), pick 2 cards at random. Probability same value?
   • (iv) Pick 5 cards with replacement from 84-card deck. Let $$ X_B $$: #black, $$ X_R $$: #red. Find distributions.
   • (v) Let $$ X = X_B - X_R $$. Compute $$ \mathbb{E}[X], \operatorname{Var}(X) $$.
   • (vi) Pick $$ n=7000 $$ cards with replacement, count $$ X_{JQK} $$. Estimate $$ \mathbb{P}(X_{JQK} \geq 2100) $$ using CLT.

Concepts & Hierarchical Analysis

1. Combinatorics and Basic Probability

• Sample Space: All possible orderings/drawings (with/without replacement).
• Event Definition: "Same value" for a pair → count pairs by value, deck size.

2. Hypergeometric & Binomial Distributions

• With Replacement: Binomial distribution for counting black/red/JQK cards.
• Without Replacement: Hypergeometric for picking specific values from a deck.

3. Conditional Probability, Bayes’ Theorem

• Posterior probability: Reverse probabilities when an event (matched value) is given.

4. Indicator Random Variables, Expectation & Variance

• Counting Matches: Sum indicators for event "value match". Use linearity of expectation for sums.
• Variance: For independent picks, variance adds up, formulas for difference of random variables.

5. Central Limit Theorem (CLT)

• Normal Approximation: For sums/counts with large $$ n $$, use normal CDF for probability estimation.

Layered Tree Breakdown

Q1. Cards & Probabilities
 ├ Combinatorics & Counting Principles
 │   └─ Number of ways to pick pairs, suits, values
 ├ Probability Calculations
 │   └─ Probability = #favorable/#total (cases for 'same value')
 ├ Discrete Random Variables & Distributions
 │   ├─ Binomial (with replacement, count of red/black/JQK cards)
 │   └─ Hypergeometric (if picking without replacement)
 ├ Conditional Probability & Bayes’ Rule
 │   └─ Reverse conditional given outcome (compute deck probabilities after event)
 ├ Expectation & Variance of Sums
 │   ├─ E[X_B], E[X_R], Var[X_B], Var[X_R], E[X_B - X_R], Var[X_B - X_R]
 │   └─ Properties of expectation/variance for independent trials
 ├ Central Limit Theorem
 │   └─ Normal approximation for sum of indicator variables; CDF Φ(z)

Q2. Joint Continuous Random Variables: Uniform Distribution on a Diamond (Geometry, Expectation, Covariance)

Restatement

2. Random variables $$X,Y$$ with joint density $$f_{(X,Y)}(x,y) = c \cdot \mathbb{1}_{|x|+|y|<1}$$.
   • (i) Determine $$c$$
   • (ii) Find marginal pdf of $$X$$.
   • (iii) Compute $$\mathbb{E}[X]$$
   • (iv) Compute $$\operatorname{Var}(X)$$
   • (v) Compute $$\operatorname{Cov}(X,Y)$$, are $$X,Y$$ independent?
   • (vi) Compute $$\mathbb{E}[X^2 Y^2]$$

Concepts & Hierarchical Analysis

1. Joint & Marginal Probability Densities

• Density Function: Uniform over region (diamond/lozenge).
• Integration: Find c so total probability is 1.

2. Marginals & Independence

• Marginalization: Integrate joint pdf over y (or x).
• Independence: If joint pdf factors, variables are independent.

3. Expectation, Variance, Covariance

• Expectation: $$\mathbb{E}[X] = \int x f_X(x) dx$$
• Variance, Covariance: Compute mix products with marginal/joint pdf.

Layered Tree Breakdown

Q2. Joint Continuous RVs (Diamond Region)
 ├ Joint PDF Normalization
 │   └─ Integrate over region |x|+|y|<1; set ∫∫=1, solve for c
 ├ Marginal Distributions
 │   └─ Integrate out y: f_X(x) = ∫ f_(X,Y)(x,y) dy
 ├ Expectation & Variance
 │   ├─ E[X] = ∫ x f_X(x) dx
 │   └─ Var[X] = ∫ (x - E[X])² f_X(x) dx
 ├ Covariance
 │   ├─ Cov(X,Y) = E[XY] - E[X]E[Y]
 │   └─ Test for independence: f_(X,Y) = f_X(x)f_Y(y)?
 ├ Higher Moments
 │   └─ E[X²Y²] = ∫∫ x²y² f_(X,Y)(x,y) dx dy

Q3. Random Subsets & Binomial-type Problems (Set Theory, Discrete Probability)

Restatement

3. Pick subset $$A$$ uniformly at random from set $$I_n = {1,\dots,n}$$. Let $$X = |A|$$, $$Y=|I_n \setminus A|$$.
   • (i) Compute $$\mathbb{P}(X=0)$$, $$\mathbb{P}(X=1)$$
   • (ii) For $$n=4$$: draw cdf of $$X$$
   • (iii) For general $$n$$: compute $$\mathbb{E}[X], \operatorname{Var}(X)$$
   • (iv) Are $${X=0}$$, $${Y=0}$$ independent?
   • (v) Does $$Y$$ have same distribution as $$X$$?
   • (vi) Compute $$\operatorname{Cov}(X,Y)$$

Concepts & Hierarchical Analysis

1. Counting & Uniform Probability on Power Set

• Sample Space: All $$2^n$$ subsets; each equally likely.
• Binomial Trait: Size of subset is binomially distributed.

2. Discrete Distributions and Generating Functions

• CDF Construction: For small $$n$$, tabulate probabilities.
• Binomial Moments: $$E[X]=np$$, $$Var[X]=np(1-p)$$ (here p=1/2).

3. Independence & Symmetry

• Events and Dependence: Test by multiplication and intersection of events.
• Symmetry: Distribution of $$X$$ and $$Y$$ due to complement; expect same.

Layered Tree Breakdown

Q3. Random Subsets, Binomial-Type
 ├ Uniform Distribution on Power Set
 │   ├─ Total outcomes: 2^n
 │   └─ Probability of given |A|: binomial coefficients
 ├ Binomial Distributions (p=1/2)
 │   ├─ X: size of random subset → Binomial(n,1/2)
 │   └─ Compute probabilities, expectation, variance
 ├ CDF Table/Distribution Graph
 ├ Independence and Symmetry
 │   └─ Use definition and compare P(X=0), P(Y=0), P(X=0 ∩ Y=0)
 ├ Covariance/Joint Properties
 │   └─ Use covariance formula, properties of complementary events

Q4. Parameter Estimation for a Family of PDFs (Moments, MLE, Bias)

Restatement

4. For random variable $$X$$ with density $$f_X(x) = \frac{3}{(x-\theta+1)^4} \mathbb{1}_{[\theta, \infty)}(x)$$:
   • (i) Compute $$\mathbb{E}[X]$$.
   • (ii) Compute $$\operatorname{Var}(X)$$.
   • (iii) For iid sample, find estimator for $$\theta$$ by method of moments.
   • (iv) Compute bias of that estimator.
   • (v) Find MLE of $$\theta$$, state bias.

Concepts & Hierarchical Analysis

1. Integration of Nonstandard PDFs

• Support: Recognize variable must be $$\geq \theta$$.
• Integrals: Use substitution for expectation/integral evaluation.

2. Method of Moments

• Match sample moment to theoretical mean; solve for parameter.

3. Maximum Likelihood Estimation

• Likelihood Function: Write for sample, maximize wrt $$\theta$$.
• Estimator Bias: $$E[\hat{\theta}]$$ vs $$\theta$$.

Layered Tree Breakdown

Q4. Estimation for Nonstandard PDF
 ├ Continuous RV Integration
 │   ├─ Compute mean, variance by integrating with given pdf/support
 |\-- Substitution method for integrals (change to known integral)
 ├ Method of Moments
 │   ├─ Set sample mean = theoretical mean, solve for parameter
 ├ Maximum Likelihood Estimation (MLE)
 │   ├─ Construct and maximize likelihood
 │   └─ Check estimator bias (unbiased/biased)

This step-by-step “mindmap” organizes the solution path for each exam question, mapping them to the relevant foundational probability/statistics concepts and the layered logical steps used to solve them. Each root node is a problem; each branch is a theory, method, or formula that underpins a solution.