Assignment 1

Author

Affiliation

Aditya Mahajan

McGill University

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

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 \(X\).

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.

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 \(\EXP[X_1 \mid Y]\).

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.