Agenda

<aside> 📌 Reminders

• First term test is October 5, 2024 </aside>

Indicator Random Variables (Indicator RVs)

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Indicator Random Variable: a special type of discrete RV that takes the value 1 if a particular event occurs, and 0 otherwise. For an event $A$, the indicator RV $I_A$ is defined as:

$$ I_A = \begin{cases} 1, & \text{if } A \text{ occurs} \\ 0, & \text{if } A \text{ does not occur} \end{cases} $$

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Example

Let $A$ be the event that the outcome of a die roll is 4.

• $I_A = 1$ if a 4 is rolled, and $I_A = 0$ otherwise.

Probability Mass Function (PMF)

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Probability Mass Function (PMF): of a discrete random variable describes the probability that the RV takes each possible value. It maps each value $x$ to a probability $P(X = x)$.

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Example

Let $X$ be the number of heads in two flips of a fair coin. The PMF of $X$ is:

$$ P(X = 0) = \frac{1}{4}, \quad P(X = 1) = \frac{1}{2}, \quad P(X = 2) = \frac{1}{4}  $$

Joint Probability Mass Function (Joint PMF)

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Joint PMF: describes the probability distribution of two or more discrete random variables simultaneously. It gives the probability that the variables take specific combinations of values. For two discrete RVs $X$ and $Y$, the joint PMF is: $p_{X, Y} (x, y) = P(X = x, Y = y)$.

This represents the probability that $X$ takes the value $x$ and $Y$ takes the value $y$.

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Properties:

• $p_{X, Y}(x, y) \geq 0$
• The sum of all joint probabilities is 1: $\sum_x \sum_y p_{X, Y}(x, y) = 1$

Example:

Let $X$ be the outcome of the first roll of a die and $Y$ the outcome of the second roll. If the dice are fair, the joint PMF for any $(x, y)$ is:

$$ P(X = x, Y = y) = \frac{1}{36}, \quad \text{for each } x, y \in \{1, 2, 3, 4, 5, 6\} $$

Midterm Practice