# Lecture 8: Random variables

## Random variables

Definition: A (real-valued) random variable $$X$$ is just a function $$X : S → ℝ$$.

Example: Suppose I roll a fair 6-sided die. On an even roll, I win \$10. On an odd roll, I lose however much money is shown. We can model the experiment (rolling a die) using the sample space $$S = \{1,2,3,4,5,6\}$$ and an equiprobable measure. The result of the experiment is given by the random variable $$X : S → \mathbb{R}$$ given by $$X(1) ::= -1$$, $$X(2) ::= 10$$, $$X(3) ::= -3$$, $$X(4) ::= 10$$, $$X(5) ::= -5$$, and $$X(6) ::= 10$$.

Definition: Given a random variable $$X$$ and a real number $$x$$, the poorly-named event $$(X = x)$$ is defined by $$(X = x) ::= \{k \in S \mid X(k) = x\}$$.

This definition is useful because it allows to ask "what is the probability that $$X = x$$?"

Definition: The probability mass function (PMF) of $$X$$ is the function $$PMF_X : \mathbb{R} → \mathbb{R}$$ given by $$PMF_X(x) = Pr(X = x)$$.

## Combining random variables

Given random variables $$X$$ and $$Y$$ on a sample space $$S$$, we can combine apply any of the normal operations of real numbers on $$X$$ and $$Y$$ by performing them pointwise on the outputs of $$X$$ and $$Y$$. For example, we can define $$X + Y : S → \mathbb{R}$$ by $$(X+Y)(k) ::= X(k) + Y(k)$$. Similarly, we can define $$X^2 : S → \mathbb{R}$$ by $$(X^2)(k) ::= \left(X(k)\right)^2$$.

We can also consider a real number $$c$$ as a random variable by defining $$C : S → \mathbb{R}$$ by $$C(k) ::= c$$. We will use the same variable for both the constant random variable and for the number itself; it should be clear from context which we are referring to.

## Indicator variables

We often want to count how many times something happens in an experiment.

Example: Suppose I flip a coin 100 times. The sample space would consist of sequences of 100 flips, and I might define the variable $$N$$ to be the number of heads. For example, $$N(H,H,H,H,\dots,H) = 100$$, while $$N(H,T,H,T,\dots) = 50$$.

A useful tool for counting is an indicator variable:

Definition: The indicator variable for an event $$A$$ is a variable having value 1 if the $$A$$ happens, and 0 otherwise.

The number of times something happens can be written as a sum of indicator variables.

In the coin example, we could define an indicator variable $$I_1$$ which is 1 if the first coin is a head, and 0 otherwise (e.g. $$I_1(H,H,H,\dots) = I_1(H,T,H,T,\dots) = 1$$). We could define a variable $$I_2$$ that only looks at the second toss, and so on. Then $$N$$ as defined above can be written as $$N = \sum I_i$$. This is useful because (as we'll see when we talk about expectation) it is often easier to reason about a sum of simple variables (like $$I_i$$) than it is to reason about a complex variable like $$N$$.

## Joint PMF of two random variables

We can summarize the probability distribution of two random variables $$X$$ and $$Y$$ using a "joint PMF". The joint PMF of $$X$$ and $$Y$$ is a function from $$\mathbb{R} \times \mathbb{R} → \mathbb{R}$$ and gives for any $$x$$ and $$y$$, the probability that $$X = x$$ and $$Y = y$$. It is often useful to draw a table:

$$Pr$$ y
1 10
x 1 1/3 1/6
10 1/6 1/3

Note that the sum of the entries in the table must be one (Exercise: prove this). You can also check that summing the rows gives the PMF of $$Y$$, while summing the columns gives the PMF of $$X$$.

## Expectation

The "expected value" is an estimate of the "likely outcome" of a random variable. It is the weighted average of all of the possible values of the RV, weighted by the probability of seeing those outcomes. Formally:

Definition: The expected value of $$X$$, written $$E(X)$$ is given by $E(X) ::= \sum_{k \in S} X(k)Pr(\{k\})$

Claim: (alternate definition of $$E(X)$$) $E(X) = \sum_{x \in \mathbb{R}} x\cdot Pr(X=x)$

Proof sketch: this is just grouping together the terms in the original definition for the outcomes with the same $$X$$ value.

Note: You may be concerned about "$$\sum_{x \in \mathbb{R}}$$. In discrete examples, $$Pr(X = x) = 0$$ almost everywhere, so this sum reduces to a finite or at least countable sum. In non-discrete example, this summation can be replaced by an integral. Measure theory is a branch of mathematics that puts this distinction on firmer theoretical footing by replacing both the summation and the integral with the so-called "Lebesgue integral". In this course, we will simply use "$$\sum$$" with the understanding that it becomes an integral when the random variable is continuous.