Probability Review

We provide a review of probability concepts. Some of the review materials have been adapted from CS229 Probability Notes and STATS310 Probability Theory Notes.

1. Elements of probability

We begin with a few basic elements of probability to establish the definition of probabilities on sets.

1.1 Probability Spaces

The probability space (\(\Omega, \mathcal{F}, \mathbf{P}\)) is a fundamental framework for expressing a random process. The sample space \(\Omega\) is the set of all the outcomes of a random experiment. Here, each outcome \(\omega \in \Omega\) can be thought of as a complete description of the state of the real world at the end of the experiment. The event space \(\mathcal{F}\) is a subset of the set of all possible sets of outcomes. It represents the collection of subsets of possible interest to us, where we denote elements of \(\mathcal{F}\) as events. The mapping \(\mathbf{P}\) assigns probabilities to each event \(A \in \mathcal{F}\). For a probability space, \(\mathcal{F}\) is furthermore a \(\sigma\)-algebra, and \(\mathbf{P}\) is a probability measure. We discuss properties of \(\sigma\)-algebras and probability measures below.

\(\sigma\)-algebra: Let \(2^{\Omega}\) denote the power set of \(\Omega\). We call \(\mathcal{F} \subseteq 2^{\Omega}\) a \(\sigma\)-algebra if

• \(\Omega \in \mathcal{F}\),
• (Closed under complement) if \(A \in\mathcal{F}\), then \(A^C \in\mathcal{F}\) (where \(A^C = \Omega \backslash A\) is the complement of \(A\)),
• (Closed under countable unions) if \(A_i \in \mathcal{F}\) for \(i = 1, 2, 3, \dots,\), then \(\bigcup_i A_i \in \mathcal{F}\).

A pair \((\Omega, \mathcal{F})\) where \(\mathcal{F}\) is a \(\sigma\)-algebra is called a measurable space.

Probability measure: Given a measurable space \((\Omega, \mathcal{F})\), a measure \(\mu\) is any set function \(\mu: \mathcal{F} \rightarrow [0, \infty]\) that satisfies the following properties:

• \(\mu(A) \ge \mu(\emptyset) = 0\) for all \(A \in \mathcal{F}\).
• \(\mu\big(\bigcup_i A_i \big) = \sum_i \mu(A_i)\) for any countable collection of disjoint sets \(A_i \in \mathcal{F}\).

When both of the above properties are satisfied and \(\mu(\Omega) = 1\), we call \(\mu\) a probability measure and denote it as \(\mathbf{P}\).

Example: Consider tossing a six-sided die. The sample space is \(\Omega = \{1, 2, 3, 4, 5, 6\}\). We can define different event spaces and probability measures on this sample space. For example, the simplest event space is the trivial event space \(\mathcal{F} = \{\emptyset, \Omega\}\). Note that this \(\mathcal{F}\) is a \(\sigma\)-algebra, as \(\emptyset\) and \(\Omega\) are complements of each other. The unique probability measure for this \(\mathcal{F}\) satisfying the requirements above is given by \(P(\emptyset) = 0\), \(P(\Omega) = 1\). Another event space is the set of all subsets of \(\Omega\). We can construct a valid probability measure for this \(\mathcal{F}\) by assigning the probability of each set in the event space to be \(\frac{i}{6}\) where \(i\) is the number of elements of that set; for example, \(P(\{1, 2, 3, 4\}) = \frac{4}{6}\) and \(P(\{1, 2, 3\}) = \frac{3}{6}\). Intuitively, this probability measure could correspond to the probability that a random fair die roll belongs to a given subset of \(\Omega\).

Additional Properties:

• \(A \subseteq B \implies P(A) \leq P(B)\).
• \(P(A \cap B) \leq \min(P(A), P(B))\).
• Union Bound: \(P(A \cup B) \leq P(A) + P(B)\).
• \(P(\Omega - A) = 1 - P(A)\).
• Law of Total Probability: If \(A_1, \dotsc, A_k\) are a set of disjoint events such that \(\bigcup^k_{i=1} A_i = \Omega\), then \(\sum^k_{i=1} P(A_i) = 1\).

1.2 Conditional probability

Let \(B\) be an event with non-zero probability. The conditional probability of any event \(A\) given \(B\) is defined as

In other words, \(P(A \mid B)\) is the probability measure of the event \(A\) after observing the occurrence of event \(B\).

1.3 Chain Rule

Let \(A_1, \dotsc, A_k\) be events, \(P(A_i) >0\). Then the chain rule states that:

Note that for \(k=2\) events, this is just the definition of conditional probability:

In general, the chain rule is derived by applying the definition of conditional probability multiple times, as in the following example:

1.4 Independence

Two events are called independent if \(P(A \cap B) = P(A)P(B)\), or equivalently, \(P(A \mid B) = P(A)\). Intuitively, \(A\) and \(B\) are independent means that observing \(B\) does not have any effect on the probability of \(A\).

2. Random variables

Oftentimes, we do not care to know the probability of a particular event. Instead, we want to know probabilities over some function of these events. For example, consider an experiment in which we flip 10 coins. Here, the elements of the sample space \(\Omega\) are 10-length sequences of heads and tails, and the event space \(\mathcal{F}\) is all subsets of \(\Omega\). We observe sequences of coin flips; for instance, \(\omega_0 = \langle H, H, T, H, T, H, H, T, T, T \rangle \in \Omega\). However, in practice we may not care to directly know the particular probability \(P(\omega_0)\) of a sequence or even the probability over a set of sequences in \(\mathcal{F}\). Instead, we might want to know the number of coins that come up heads or the length of the longest run of tails. These quantities are functions of \(\omega \in \Omega\), which we refer to as random variables.

More formally, define a mapping \(X : \Omega \to E\) between two measurable spaces \((\Omega, \mathcal{F})\) and \((E, \mathcal{E})\), where \(\mathcal{E}\) is a \(\sigma\)-algebra on \(E\). Then, \(X\) is a random variable if \(X^{-1}(B) := \{\omega: X(\omega) \in B \} \in \mathcal{F}\) for all \(B \in \mathcal{E}\). Intuitively, this means that every set \(B\) is associated with a set of outcomes that belongs to \(\mathcal{F}\) and has a well-defined probability. Typically, we denote random variables using upper case letters \(X(\omega)\) or more simply \(X\) (where the dependence on the random outcome \(\omega\) is implied). We denote the value that a random variable may take on using lower case letters \(x\). Thus, \(X = x\) denotes the event that the random variable \(X\) takes on the value \(x \in E\).

Example: In our experiment above, suppose that \(X(\omega)\) is the number of heads which occur in the sequence of tosses \(\omega\). Given that only 10 coins are tossed, \(X(\omega)\) can take only a finite number of values (0 through 10), so it is known as a discrete random variable. Here, the probability of the set associated with a random variable \(X\) taking on some specific value \(k\) is \(P(X = k) := P(\{\omega : X(\omega) = k\}) = P(\omega \in \text{all sequences with k heads})\). Note that the set of all sequences with \(k\) heads is an element of \(\mathcal{F}\), given that \(\mathcal{F}\) consists of all subsets of \(\Omega\).

Example: Suppose that \(X(\omega)\) is a random variable indicating the amount of time it takes for a radioactive particle to decay (\(\omega\) for this example could be some underlying characterization of the particle that changes as it decays). In this case, \(X(\omega)\) takes on an infinite number of possible values, so it is called a continuous random variable. We denote the probability that \(X\) takes on a value between two real constants \(a\) and \(b\) (where \(a < b\)) as \(P(a \leq X \leq b) := P(\{\omega : a \leq X(\omega) \leq b\})\).

When describing the event that a random variable takes on a certain value, we often use the indicator function \(\mathbf{1}\{A\}\) which takes value 1 when event \(A\) happens and 0 otherwise. For example, for a random variable \(X\),

In order to specify the probability measures used when dealing with random variables, it is often convenient to specify alternative functions from which the probability measure governing an experiment immediately follows. In the following three sections, we describe these functions: the cumulative distribution function (CDF), the probability mass function (PMF) for discrete random variables, and the probability density function (PDF) for continuous random variables. For the rest of this section, we suppose that \(X\) takes on real values, i.e., \(E = \mathbb{R}\).

2.1 Cumulative distribution functions

A cumulative distribution function (CDF) is a function \(F_X : \mathbb{R} \to [0, 1]\) which specifies a probability measure as

By using this function, one can calculate the probability that \(X\) takes on a value between any two real constants \(a\) and \(b\) (where \(a < b\)).

Properties:

• \(0 \leq F_X(x) \leq 1\). This follows from the definition of the probability measure.
• \(\lim_{x \to -\infty} F_X(x) = 0\). As \(x\) approaches \(-\infty\), the corresponding set of \(\omega\) where \(X(\Omega) \le x\) approaches \(\emptyset\), for which \(P(\emptyset) = 0\).
• \(\lim_{x \to +\infty} F_X(x) = 1\). As \(x\) approaches \(\infty\), the corresponding set of \(\omega\) where \(X(\omega) \le x\) approaches \(\Omega\), for which \(P(\Omega) = 1\).
• \(x \leq y \implies F_X(x) \leq F_X(y)\). This follows from the fact that the event that \(X \le x\) is a subset of \(X \le y\) for \(x \leq y\).

2.2 Probability mass functions

Suppose a random variable \(X\) takes on a finite set of possible values (i.e., \(X\) is a discrete random variable). A simpler way to represent the probability measure associated with a random variable is to directly specify the probability of each value that the random variable can assume. Let \(Val(X)\) refer to the set of possible values that the random variable \(X\) may assume; for example, if \(X(\omega)\) is a random variable indicating the number of heads out of ten tosses of coin, then \(Val(X) = \{0, 1, 2, \dotsc, 10\}\). Then, a probability mass function (PMF) is a function \(p_X : Val(X) \to [0, 1]\) such that \(p_X(x) = P(X = x)\).

Properties:

• \(0 \leq p_X(x) \leq 1\).
• \(\sum_{x \in A} p_X(x) = P(X \in A)\). This follows by the property that probability measures apply over countable unions of disjoint sets.
• \(\sum_{x \in Val(X)} p_X(x) = 1\). Applying the previous property, we have that \(\sum_{x \in Val(X)} p_X(x) = P(X \in Val(X)) = P(\Omega) = 1\).

2.3 Probability density functions

For some continuous random variables, the cumulative distribution function \(F_X(x)\) is differentiable everywhere. In these cases, we define the probability density function (PDF) as the derivative of the CDF, i.e.,

Note that the PDF for a continuous random variable may not always exist (i.e., if \(F_X(x)\) is not differentiable everywhere).

According to the properties of differentiation, for very small \(\delta x\),

Both CDFs and PDFs (when they exist) can be used for calculating the probabilities of different events. But it should be emphasized that the value of PDF at any given point \(x\) is not the probability of that event, i.e., \(f_X(x) \neq P(X = x)\). Because \(X\) can take on infinitely many values, it holds that \(P(X = x) = 0\). On the other hand, \(f_X(x)\) can take on values larger than one (but the integral of \(f_X(x)\) over any subset of \(\Re\) will be at most one).

Properties:

• \(f_X(x) \geq 0\).
• \(\int^{\infty}_{-\infty} f_X(x) dx = 1\).
• \(\int_{x \in A} f_X(x) dx = P(X \in A)\).

2.4 Expectation

Suppose that \(X\) is a discrete random variable with PMF \(p_X(x)\) and \(g : \Re \to \Re\) is an arbitrary function. In this case, \(g(X)\) can be considered a random variable, and we define the expectation or expected value of \(g(X)\) as

If \(X\) is a continuous random variable with PDF \(f_X(x)\), then the expected value of g(X) is defined as

Intuitively, the expectation of \(g(X)\) can be thought of as a “weighted average” of the values that \(g(x)\) can take on, where the weights are given by \(p_X(x)\) or \(f_X(x)\) which add up to \(1\) over all \(x\). As a special case of the above, note that the expectation, \(\E[X]\) of a random variable itself is found by letting \(g(x) = x\); this is also known as the mean of the random variable \(X\).

Properties:

• \(\E[a] = a\) for any constant \(a \in \Re\).
• \(\E[af(X)] = a\E[f(X)]\) for any constant \(a \in \Re\).
• (Linearity of Expectation) \(\E[f(X) + g(X)] = \E[f(X)] + \E[g(X)]\).
• For a discrete random variable \(X\), \(\E[\mathbf{1}\{X = k\}] = P(X = k)\).

2.5 Variance

The variance of a random variable \(X\) is a measure of how concentrated the distribution of a random variable \(X\) is around its mean. Formally, the variance of a random variable \(X\) is defined as \(Var[X] = \E[(X - \E[X])^2]\).

Using the properties in the previous section, we can derive an alternate expression for the variance:

where the second equality follows from linearity of expectations and the fact that \(\E[X]\) is actually a constant with respect to the outer expectation.

Properties:

• \(Var[a] = 0\) for any constant \(a \in \Re\).
• \(Var[af(X)] = a^2 Var[f(X)]\) for any constant \(a \in \Re\).

Example: Calculate the mean and the variance of the uniform random variable \(X\) with PDF \(f_X(x) = 1, \forall x \in [0, 1], 0\) elsewhere.

Example: Suppose that \(g(x) = \mathbf{1}\{x \in A\}\) for some subset \(A \subseteq \Omega\). What is \(\E[g(X)]\)?

• Discrete case:

• Continuous case:

2.6 Some common random variables

Discrete random variables

• \(X \sim \text{Bernoulli}(p)\) (where \(0 \leq p \leq 1\)): the outcome of a coin flip (\(H=1, T=0\)) for a coin that comes up heads with probability \(p\).

• \(X \sim \text{Binomial}(n, p)\) (where \(0 \leq p \leq 1\)): the number of heads in \(n\) independent flips of a coin with heads probability \(p\).

• \(X \sim \text{Geometric}(p)\) (where \(p > 0\)): the number of flips of a coin until the first heads, for a coin that comes up heads with probability \(p\).

• \(X \sim \text{Poisson}(\lambda)\) (where \(\lambda\) > 0): a probability distribution over the nonnegative integers used for modeling the frequency of rare events.

Continuous random variables

• \(X \sim \text{Uniform}(a, b)\) (where \(a < b\)): equal probability density to every value between \(a\) and \(b\) on the real line.

• \(X \sim \text{Exponential}(\lambda)\) (where \(\lambda\) > 0): decaying probability density over the nonnegative reals.

• \(X \sim \text{Normal}(\mu, \sigma^2)\): also known as the Gaussian distribution

3. Two random variables

Thus far, we have considered single random variables. In many situations, however, there may be more than one quantity that we are interested in knowing during a random experiment. For instance, in an experiment where we flip a coin ten times, we may care about both \(X(\omega) =\) the number of heads that come up as well as \(Y(\omega) =\) the length of the longest run of consecutive heads. In this section, we consider the setting of two random variables. We first discuss joint and marginal CDFs, then joint and marginal PMFs and PDFs.

3.1 Joint and marginal cumulative distribution functions

Suppose that we have two random variables \(X\) and \(Y\). One way to work with these two random variables is to consider each of them separately. If we do that we will need the CDFs \(F_X(x)\) and \(F_Y (y)\). But if we want to know about the values that \(X\) and \(Y\) assume simultaneously during outcomes of a random experiment, we require a more complicated structure known as the joint cumulative distribution function of \(X\) and \(Y\), defined by

It can be shown that by knowing the joint cumulative distribution function, the probability of any event involving \(X\) and \(Y\) can be calculated.

The joint CDF \(F_{XY} (x, y)\) and the cumulative distribution functions \(F_X(x)\) and \(F_Y (y)\) of each variable separately are related by

Here, we call \(F_X(x)\) and \(F_Y(y)\) the marginal cumulative distribution functions of \(F_{XY} (x, y)\).

Properties:

The properties of CDFs on a single random variable also translate over to joint cumulative distribution functions.

• \(0 \leq F_{XY} (x, y) \leq 1\).
• \(\lim_{x,y\to \infty} F_{XY} (x, y) = 1\).
• \(\lim_{x,y\to -\infty} F_{XY} (x, y) = 0\).

3.2 Joint and marginal probability mass functions

If \(X\) and \(Y\) are discrete random variables, then the joint probability mass function \(p_{XY} : Val(X) \times Val(Y) \to [0, 1]\) is defined by

Here, \(0 \leq p_{XY}(x, y) \leq 1\) for all \(x, y,\) and \(\sum_{x \in Val(X)} \sum_{y \in Val(Y)} p_{XY}(x, y) = 1\).

How does the joint PMF over two variables relate to the probability mass function for each variable separately? It turns out that

and similarly for \(p_Y(y)\). In this case, we refer to \(p_X(x)\) as the marginal probability mass function of \(X\). In statistics, the process of forming the marginal distribution with respect to one variable by summing out the other variable is often known as “marginalization.”

3.3 Joint and marginal probability density functions

Let \(X\) and \(Y\) be two continuous random variables with joint cumulative distribution function \(F_{XY}\). In the case that \(F_{XY}(x, y)\) is everywhere differentiable in both \(x\) and \(y\), then we can define the joint probability density function,

Like in the single-dimensional case, \(f_{XY} (x, y) \neq P(X = x, Y = y)\), but rather

Note that the values of the probability density function \(f_{XY}(x, y)\) are always nonnegative, but they may be greater than 1. Nonetheless, it must be the case that \(\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} f_{XY}(x,y) = 1\).

Analagous to the discrete case, we define

as the marginal probability density function (or marginal density) of \(X\), and similarly for \(f_Y (y)\).

3.4 Conditional distributions

Conditional distributions seek to answer the question, what is the probability distribution over \(Y\), when we know that \(X\) must take on a certain value \(x\)? In the discrete case, the conditional probability mass function of \(Y\) given \(X\) is simply

assuming that \(p_X(x) \neq 0\).

In the continuous case, the situation is technically a little more complicated because the probability that a continuous random variable \(X\) takes on a specific value \(x\) is equal to zero. Ignoring this technical point, we simply define, by analogy to the discrete case, the conditional probability density of \(Y\) given \(X = x\) to be

\(f_{Y \mid X}(y \mid x) = \frac{f_{XY} (x, y)}{f_X(x)}\),

provided \(f_X(x) \neq 0\).

3.5 Chain rule

The chain rule we derived earlier in Section 1.3 for events can be applied to random variables as follows:

3.6 Bayes’ rule

A useful formula that often arises when trying to derive expressions for conditional probability is Bayes’ rule, which arises from applying the chain rule:

If the random variables \(X\) and \(Y\) are continuous,

3.7 Independence

Two random variables \(X\) and \(Y\) are independent if \(F_{XY} (x, y) = F_X(x)F_Y(y)\) for all values of \(x\) and \(y\). Equivalently,

• For discrete random variables, \(p_{XY} (x, y) = p_X(x)p_Y(y)\) for all \(x \in Val(X)\), \(y \in Val(Y)\).
• For discrete random variables, \(p_{Y\mid X}(y \mid x) = p_Y(y)\) whenever \(p_X(x) \neq 0\) for all \(y \in Val(Y)\).
• For continuous random variables, \(f_{XY} (x, y) = f_X(x)f_Y(y)\) for all \(x, y \in \Re\).
• For continuous random variables, \(f_{Y\mid X}(y \mid x) = f_Y(y)\) whenever \(f_X(x) \neq 0\) for all \(y \in \Re\).

Informally, two random variables \(X\) and \(Y\) are independent if “knowing” the value of one variable will never have any effect on the conditional probability distribution of the other variable, that is, you know all the information about the pair \((X, Y)\) by just knowing \(f(x)\) and \(f(y)\). The following lemma formalizes this observation:

Lemma 3.1. If \(X\) and \(Y\) are independent, then for any subsets \(A, B \subseteq \Re\), we have,

By using the above lemma, one can prove that if \(X\) is independent of \(Y\) then any function of \(X\) is independent of any function of \(Y\).

3.8 Expectation and covariance

Suppose that we have two discrete random variables \(X, Y\) and \(g : \Re^2 \to \Re\) is a function of these two random variables. Then the expected value of \(g\) is defined as

For continuous random variables \(X, Y\), the analogous expression is

We can use the concept of expectation to study the relationship of two random variables with each other. In particular, the covariance of two random variables \(X\) and \(Y\) is defined as

Intuitively, the covariance of \(X\) and \(Y\) measures how often and by how much \(X\) and \(Y\) are both greater than or less than their respective means. If larger values of \(X\) correspond with larger values of \(Y\) and vice versa, then covariance is positive. If larger values of \(X\) correspond with smaller values of \(Y\) and vice versa, then covariance is negative. Using an argument similar to that for variance, we can rewrite this as

Here, the key step in showing the equality of the two forms of covariance is in the third equality, where we use the fact that \(\E[X]\) and \(\E[Y]\) are actually constants which can be pulled out of the expectation. When \(Cov[X, Y] = 0\), we say that \(X\) and \(Y\) are uncorrelated.

Properties:

• (Linearity of expectation) \(\E[f(X, Y) + g(X, Y)] = \E[f(X, Y)] + \E[g(X, Y)]\).
• \(Var[X + Y] = Var[X] + Var[Y] + 2Cov[X, Y]\).
• If \(X\) and \(Y\) are independent, then \(Cov[X, Y] = 0\). However, if \(Cov[X, Y] = 0\), it is not necessarily true that \(X\) and \(Y\) are independent. For example, let \(X \sim \text{Uniform}(-1, 1)\) and let \(Y = X^2\). Then, \(Cov[X, Y] = \E[X^3] - \E[X]\E[X^2] = \E[X^3] - 0\cdot \E[X^2] = 0\) even though \(X\) and \(Y\) are not independent.
• If \(X\) and \(Y\) are independent, then \(\E[f(X)g(Y)] = \E[f(X)]\E[g(Y)]\).