Random variables

Defining variables

We have a sample space, \(\Omega \). A random variable \(X\) is a mapping from the sample space to the real numbers:

\(X: \Omega \rightarrow \mathbb{R}\)

We can then define the set of elements in \(\Omega \). As an example, take a coin toss and a die roll. The sample space is:


A random variable could give us just the die value, such that:


We can define this more precisely using set-builder notation, by saying the following is defined for all \(c\in \mathbb{R}\):

\(\{\omega |X(\omega )\le c\}\)

That is, for any number random variable map \(X\), there is a corresponding subset of \(\Omega \) containing the \(\omega \)s in \(\Omega \) which map to less than \(c\).

Multiple variables

Multiple variables can be defined on the sample space. If we rolled a die we could define variables for

  • Whether it was odd/even

  • Number on the die

  • Whether it was less than 3

With more die we could add even more variables

Derivative variables

If we define a variable \(X\), we can also define another variable \(Y=X^2\).

Probability mass functions

\(P(X=x)=P({\omega |X(\omega)=x})\)

For discrete probability, this is a helpful number. For example for rolling a die.

This is not helpful for continuous probability, where the chance of any specific outcome is \(0\).

Cumulative distribution functions


Random variables all valued as real numbers, and so we can write:

\(P(X\le x)=P({\omega |X(\omega)\le x})\)


\(F_X(x)=\int_{-\infty}^x f_X(u)du\)

\(F_X(x)=\sum_{x_i\le x}P(X=x_i) \)


\(P(X\le x)+P(X\ge x)-P(X=x)=1\)


\(P(a< X\le b)=F_X(b)-F_X(a)\)

Probability density functions


If continuous, probability at any point is \(0\). We instead look at probability density.

Derived from cumulative distribution function:

\(F_X(x)=\int_{-\infty}^x f_X(u)du\)

The density function is \(f_X(x)\).

Conditional probability distributions

For probability mass functions:

\(P(Y=y|X=x)=\dfrac{P(Y=y\land X=x)}{P(X=x)}\)

For probability density functions:


Multiple variables

Joint and marginal probability

Joint probability

\(P(X=x\land Y=y)\)

Marginal probability

\(P(X=x)=\sum_{y}P(X=x\land Y=y)\)


Independence and conditional independence


\(x\) is independent of \(y\) if:

\(\forall x_i \in x,\forall y_j \in y (P(x_i|y_j)=P(x_i)\)

If \(P(x_i|y_j)=P(x_i)\) then:

\(P(x_i\land y_j)=P(x_i).P(y_j)\)

This logic extends beyond just two events. If the events are independent then:

\(P(x_i\land y_j \land z_j)=P(x_i).P(y_j \land z_k)=P(x_i).P(y_j).P(z_k)\)

Note that because:

\(P(x_i|y_j)=\dfrac{P(x_i\land y_j)}{P(y_j)}\)

If two variables are independent



Conditional independence

\(P(A\land B|X)=P(A|X)P(B|X)\)

This is the same as:

\(P(A|B\land X)=P(A|X)\)