Agenda

<aside> 📌 Reminders

• Quiz 2 released Friday, due September 22, 2024 </aside>

Introducing Random Variables & PMFs

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Random Variable: a numerical quantity that depends on the outcome of an event

$X(\text{outcome}) \in \mathbb{R}$

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For example, $X$ could represent the difference between the rolls of a die, $X((\text{roll}_1, \text{roll}_2) = \text{roll}_1 - \text{roll}_2$

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Probability Mass Function (PMF): assigns a probability that a (discrete) random variable takes on a particular value; typically denoted $p_X(x)$ where $X$ is the random variable and $x$ is the value it takes on, i.e. $p_X(x) = P(X = x)$

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For example, using our RV from before, $p_X(x) = P(X = x) = P(\text{roll}_1 - \text{roll}_2 = x)$

Some properties of PMF:

1. $\forall x, 0 \leq p_X(x) \leq 1$
2. $\sum_x p_X(x) = 1$

Practice with RVs and PMFs

Independence

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Conditional Probability: The probability of some event $A$ occurring given that some other event $B$ has occurred, expressed as $P(A|B) = \frac{P(A \cap B)}{P(B)}$

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Independence: Two events $A$ and $B$ are independent if the occurrence of one has no affect on the other, i.e. $P(A|B) = P(A)$

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• A consequence of this is that for independent events $P(A \cap B) = P(A)P(B)$, why?

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Conditional Independence: Two events $A$ and $B$ are conditionally independent given some third event $C$ if, given the occurrence of $C$, the occurrence of $B$ has not affect on $A$ and vice versa i.e. $P(A | B \cap C) = P(A|C)$

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