Assignment 1

Author

Affiliation

Aditya Mahajan

McGill University

The purpose of this assignment is to review the background knowledge needed for this course.

Question 1 A fair coin is tossed twice, where the two tosses are independent. Let \(Ω = \{ \mathsf{H}, \mathsf{T} \} × \{ \mathsf{H}, \mathsf{T} \}\) denote the sample space.

1. Describe the \(σ\)-algebra \(\ALPHABET F\) and a probability function \(\PR \colon \ALPHABET F \to [0,1]\) such that \((Ω, \ALPHABET F, \PR)\) is a probability space.

2. Verify that \(\PR\) satisfies the axioms of probability.

3. Let \(X\) be a random variable that counts the number of heads. Find the probability mass function and the cumulative density function of \(X\).

Question 2 Suppose \(X\) and \(Y\) are random variables that are uniformly distributed in the shaded region shown in Figure 1. Compute \[ \PR( X - Y < 1 \mid Y < 2). \]

f_X_Y_2025_mid_term_1 = [
  {x: 0, y: 0},
  {x: 4, y: 0},
  {x: 4, y: 4},
  {x: 0, y: 0},
];

f_X_Y_2025_mid_term_1_A = [
  {x: 0, y: 0},
  {x: 4, y: 4},
];

f_X_Y_2025_mid_term_1_A_2 = [
  {x: 1, y: 0},
  {x: 4, y: 3},
];

f_X_Y_2025_mid_term_1_B = [
  {x: 0, y: 0},
  {x: 4, y: 0},
  {x: 4, y: 2},
  {x: 2, y: 2},
  {x: 0, y: 0},
];

Plot.plot({
  grid: true,
  width: 300,
  height: 300,
  marks: [
    // Axes
    Plot.ruleX([0]),
    Plot.ruleY([0]),
    // Data
    Plot.areaY(f_X_Y_2025_mid_term_1, {x: "x", y: "y", fill: "lightblue"}),
    Plot.line(f_X_Y_2025_mid_term_1, {x: "x", y: "y", stroke: "black"}),
  ]
})

Figure 1: The joint PDF \(f_{X,Y}\) is uniform in the shaded region and zero outside.

Question 3 Suppose \(X_1 \sim \text{Binomial}(2, \tfrac 12)\) and \(X_2 \sim \text{Binomial}(1, \tfrac 12)\) are independent random variables, and \(Y = X_1 + X_2\). Find \(\EXP[X_1 \mid Y]\).

Question 4 A fair coin is tossed repeatedly until the sequence HHT shows up. Model this as a Markov chain and compute the expected number of coin tosses until stopping.