AHL 4.14—Properties of discrete and continuous random variables

A random variable is a variable whose value is determined by the outcome of a random experiment. At HL, we study both discrete and continuous random variables in depth, including their measures of centre and spread, and how linear transformations affect them.

Discrete Random Variable: A random variable $X$ that can take only a countable number of distinct values (e.g. 0, 1, 2, 3, ...). Each value has an associated probability $P(X=x_k)$, and the sum of all probabilities equals 1: ${\sum }_{k}P(X=x_k)=1$

Continuous Random Variable: A random variable $X$ that can take any value within a given interval (or union of intervals). It is described by a probability density function (PDF) rather than individual probabilities at each point.

Expected Value (Mean) of a Discrete Random Variable

Before studying variance, we need the expected value , the probability-weighted average of all possible values:

Expected Value of a Discrete Random Variable: $E(X)={\sum }_k{x}_{k}P(X=x_k)$
This is also called the mean and denoted $\mu$.

Mode and Median of a Discrete Random Variable

Mode (Discrete): The value of $x$ with the highest probability $P(X=x)$. It is the most likely outcome.

Median (Discrete): The value $m$ such that $P(X\le m)\ge 0.5$ and $P(X\ge m)\ge 0.5$. For small distributions, it is found by accumulating probabilities in order until the cumulative total first reaches or exceeds 0.5.

For a discrete random variable $X$ with mean $\mu =E(X)$, the variance measures how spread out the values are around the mean.

Variance of a Discrete Random Variable: $\text{Var}(X)=E\left[(X-\mu {)}^2\right]={\sum }_k(x_k-\mu {)}^{2}P(X=x_k)$

In practice, the following equivalent formula is far more efficient for calculation:

$\boxed{\text{Var}(X)=E(X^2)-[E(X){]}^2}$

where $E(X^2)={\sum }_k{x}_k^{2}P(X=x_k)$.