# Chapter 2 Probability Review

We give a very brief review of some necessary probability concepts. As the treatment is less than complete, a list of references is given at the end of the chapter. For example, we ignore the usual recap of basic set theory and omit proofs and examples.

## 2.1 Probability Models

When discussing probability models, we speak of random experiments that produce one of a number of possible outcomes.

A probability model that describes the uncertainty of an experiment consists of two elements:

• The sample space, often denoted as $$\Omega$$, which is a set that contains all possible outcomes.
• A probability function that assigns to an event $$A$$ a nonnegative number, $$P[A]$$, that represents how likely it is that event $$A$$ occurs as a result of the experiment.

We call $$P[A]$$ the probability of event $$A$$. An event $$A$$ could be any subset of the sample space, not necessarily a single possible outcome. The probability law must follow a number of rules, which are the result of a set of axioms that we introduce now.

## 2.2 Probability Axioms

Given a sample space $$\Omega$$ for a particular experiment, the probability function associated with the experiment must satisfy the following axioms.

1. Nonnegativity: $$P[A] \geq 0$$ for any event $$A \subset \Omega$$.
2. Normalization: $$P[\Omega] = 1$$. That is, the probability of the entire space is 1.
3. Additivity: For mutually exclusive events $$E_1, E_2, \ldots$$ $P\left[\bigcup_{i = 1}^{\infty} E_i\right] = \sum_{i = 1}^{\infty} P[E_i]$

Using these axioms, many additional probability rules can easily be derived.

## 2.3 Probability Rules

Given an event $$A$$, and its complement, $$A^c$$, that is, the outcomes in $$\Omega$$ which are not in $$A$$, we have the complement rule:

$P[A^c] = 1 - P[A]$

In general, for two events $$A$$ and $$B$$, we have the addition rule:

$P[A \cup B] = P[A] + P[B] - P[A \cap B]$

If $$A$$ and $$B$$ are also disjoint, then we have:

$P[A \cup B] = P[A] + P[B]$

If we have $$n$$ mutually exclusive events, $$E_1, E_2, \ldots E_n$$, then we have:

$P\left[\textstyle\bigcup_{i = 1}^{n} E_i\right] = \sum_{i = 1}^{n} P[E_i]$

Often, we would like to understand the probability of an event $$A$$, given some information about the outcome of event $$B$$. In that case, we have the conditional probability rule provided $$P[B] > 0$$.

$P[A \mid B] = \frac{P[A \cap B]}{P[B]}$

Rearranging the conditional probability rule, we obtain the multiplication rule:

$P[A \cap B] = P[B] \cdot P[A \mid B] \cdot$

For a number of events $$E_1, E_2, \ldots E_n$$, the multiplication rule can be expanded into the chain rule:

$P\left[\textstyle\bigcap_{i = 1}^{n} E_i\right] = P[E_1] \cdot P[E_2 \mid E_1] \cdot P[E_3 \mid E_1 \cap E_2] \cdots P\left[E_n \mid \textstyle\bigcap_{i = 1}^{n - 1} E_i\right]$

Define a partition of a sample space $$\Omega$$ to be a set of disjoint events $$A_1, A_2, \ldots, A_n$$ whose union is the sample space $$\Omega$$. That is

$A_i \cap A_j = \emptyset$

for all $$i \neq j$$, and

$\bigcup_{i = 1}^{n} A_i = \Omega.$

Now, let $$A_1, A_2, \ldots, A_n$$ form a partition of the sample space where $$P[A_i] > 0$$ for all $$i$$. Then for any event $$B$$ with $$P[B] > 0$$ we have Bayes’ Rule:

$P[A_i | B] = \frac{P[A_i]P[B | A_i]}{P[B]} = \frac{P[A_i]P[B | A_i]}{\sum_{i = 1}^{n}P[A_i]P[B | A_i]}$

The denominator of the latter equality is often called the law of total probability:

$P[B] = \sum_{i = 1}^{n}P[A_i]P[B | A_i]$

Two events $$A$$ and $$B$$ are said to be independent if they satisfy

$P[A \cap B] = P[A] \cdot P[B]$

This becomes the new multiplication rule for independent events.

A collection of events $$E_1, E_2, \ldots E_n$$ is said to be independent if

$P\left[\bigcap_{i \in S} E_i \right] = \prod_{i \in S}P[E_i]$

for every subset $$S$$ of $$\{1, 2, \ldots n\}$$.

If this is the case, then the chain rule is greatly simplified to:

$P\left[\textstyle\bigcap_{i = 1}^{n} E_i\right] = \prod_{i=1}^{n}P[E_i]$

## 2.4 Random Variables

A random variable is simply a function which maps outcomes in the sample space to real numbers.

### 2.4.1 Distributions

We often talk about the distribution of a random variable, which can be thought of as:

$\text{distribution} = \text{list of possible} \textbf{ values} + \text{associated} \textbf{ probabilities}$

This is not a strict mathematical definition, but is useful for conveying the idea.

If the possible values of a random variables are discrete, it is called a discrete random variable. If the possible values of a random variables are continuous, it is called a continuous random variable.

### 2.4.2 Discrete Random Variables

The distribution of a discrete random variable $$X$$ is most often specified by a list of possible values and a probability mass function, $$p(x)$$. The mass function directly gives probabilities, that is,

$p(x) = p_X(x) = P[X = x].$

Note we almost always drop the subscript from the more correct $$p_X(x)$$ and simply refer to $$p(x)$$. The relevant random variable is discerned from context

The most common example of a discrete random variable is a binomial random variable. The mass function of a binomial random variable $$X$$, is given by

$p(x | n, p) = {n \choose x} p^x(1 - p)^{n - x}, \ \ \ x = 0, 1, \ldots, n, \ n \in \mathbb{N}, \ 0 < p < 1.$

This line conveys a large amount of information.

• The function $$p(x | n, p)$$ is the mass function. It is a function of $$x$$, the possible values of the random variable $$X$$. It is conditional on the parameters $$n$$ and $$p$$. Different values of these parameters specify different binomial distributions.
• $$x = 0, 1, \ldots, n$$ indicates the sample space, that is, the possible values of the random variable.
• $$n \in \mathbb{N}$$ and $$0 < p < 1$$ specify the parameter spaces. These are the possible values of the parameters that give a valid binomial distribution.

Often all of this information is simply encoded by writing

$X \sim \text{bin}(n, p).$

### 2.4.3 Continuous Random Variables

The distribution of a continuous random variable $$X$$ is most often specified by a set of possible values and a probability density function, $$f(x)$$. (A cumulative density or moment generating function would also suffice.)

The probability of the event $$a < X < b$$ is calculated as

$P[a < X < b] = \int_{a}^{b} f(x)dx.$

Note that densities are not probabilities.

The most common example of a continuous random variable is a normal random variable. The density of a normal random variable $$X$$, is given by

$f(x | \mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} \cdot \exp\left[\frac{-1}{2} \left(\frac{x - \mu}{\sigma}\right)^2 \right], \ \ \ -\infty < x < \infty, \ -\infty < \mu < \infty, \ \sigma > 0.$

• The function $$f(x | \mu, \sigma^2)$$ is the density function. It is a function of $$x$$, the possible values of the random variable $$X$$. It is conditional on the paramters $$\mu$$ and $$\sigma^2$$. Different values of these parameters specify different normal distributions.
• $$-\infty < x < \infty$$ indicates the sample space. In this case, the random variable may take any value on the real line.
• $$-\infty < \mu < \infty$$ and $$\sigma > 0$$ specify the parameter space. These are the possible values of the parameters that give a valid normal distribution.

Often all of this information is simply encoded by writing

$X \sim N(\mu, \sigma^2)$

### 2.4.4 Several Random Variables

Consider two random variables $$X$$ and $$Y$$. We say they are independent if

$f(x, y) = f(x) \cdot f(y)$

for all $$x$$ and $$y$$. Here $$f(x, y)$$ is the joint density (mass) function of $$X$$ and $$Y$$. We call $$f(x)$$ the marginal density (mass) function of $$X$$. Then $$f(y)$$ the marginal density (mass) function of $$Y$$. The joint density (mass) function $$f(x, y)$$ together with the possible $$(x, y)$$ values specify the joint distribution of $$X$$ and $$Y$$.

Similar notions exist for more than two variables.

## 2.5 Expectations

For discrete random variables, we define the expectation of the function of a random variable $$X$$ as follows.

$\mathbb{E}[g(X)] \triangleq \sum_{x} g(x)p(x)$

For continuous random variables we have a similar definition.

$\mathbb{E}[g(X)] \triangleq \int g(x)f(x) dx$

For specific functions $$g$$, expectations are given names.

The mean of a random variable $$X$$ is given by

$\mu_{X} = \text{mean}[X] \triangleq \mathbb{E}[X].$

So for a discrete random variable, we would have

$\text{mean}[X] = \sum_{x} x \cdot p(x)$

For a continuous random variable we would simply replace the sum by an integral.

The variance of a random variable $$X$$ is given by

$\sigma^2_{X} = \text{var}[X] \triangleq \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2.$

The standard deviation of a random variable $$X$$ is given by

$\sigma_{X} = \text{sd}[X] \triangleq \sqrt{\sigma^2_{X}} = \sqrt{\text{var}[X]}.$

The covariance of random variables $$X$$ and $$Y$$ is given by

$\text{cov}[X, Y] \triangleq \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])] = \mathbb{E}[XY] - \mathbb{E}[X] \cdot \mathbb{E}[Y].$

## 2.6 Likelihood

Consider $$n$$ iid random variables $$X_1, X_2, \ldots X_n$$. We can then write their likelihood as

$\mathcal{L}(\theta \mid x_1, x_2, \ldots x_n) = \prod_{i = i}^n f(x_i; \theta)$

where $$f(x_i; \theta)$$ is the density (or mass) function of random variable $$X_i$$ evaluated at $$x_i$$ with parameter $$\theta$$.

Whereas a probability is a function of a possible observed value given a particular parameter value, a likelihood is the opposite. It is a function of a possible parameter value given observed data.

Maximumizing likelihood is a common techinque for fitting a model to data.

## 2.7 Videos

The YouTube channel mathematicalmonk has a great Probability Primer playlist containing lectures on many fundamental probability concepts. Some of the more important concepts are covered in the following videos:

## 2.8 References

Any of the following are either dedicated to, or contain a good coverage of the details of the topics above.