IB Mathematics AI SL Concept of discrete random variables and their probability MAI Study Notes - New Syllabus

PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE X

A random variable \( X \) takes on some values in a given domain at random!!! It may be A discrete variable takes on values in a finite or numerable set, while a continuous variable takes on values in some interval(s).

DISCRETE RANDOM VARIABLE

Let \( X \) be a variable which takes on the values
$ 10, 20, 30 $
with probabilities
$ 0.2, 0.3, 0.5 $
respectively. We often use a table:

1. All the probabilities are non-negative numbers; and
2. Their sum is always 1.

Then we say that \( X \) is a discrete random variable.

To express that the probability that \( X=10 \) is 0.2, we write:
$ P(X=10) = 0.2 $
Similarly, \( P(X=20) = 0.3 \) and \( P(X=30) = 0.5 \).

In general, for a discrete random variable \( X \) with:

$ \begin{array}{c|cccc}
X & x_1 & x_2 & x_3 & \cdots \\
\hline
P(X=x) & p_1 & p_2 & p_3 & \cdots \\
\end{array} $

it holds:
1. \( p_i \geq 0 \), for all \( i \).
2. \( \sum p_i = 1 \), i.e., \( p_1 + p_2 + p_3 + \cdots = 1 \).

We write:
$ P(X=x_1) = p_1, \quad P(X=x_2) = p_2, \quad \text{and so on.} $
(We also say that a probability function \( p: x_i \mapsto y_i \) is defined).

◆ THE EXPECTED VALUE \( \mu = E(X) \)

The mean \( \mu \) or otherwise the expected value \( E(X) \) is defined by:
$ E(X) = \sum x_i p_i = x_1 p_1 + x_2 p_2 + x_3 p_3 + \cdots $

For our example:

$ \begin{array}{c|ccc}
X & 10 & 20 & 30 \\
\hline
P(X=x) & 0.2 & 0.3 & 0.5 \\
\end{array} $

the expected value (otherwise the mean) is:
$ E(X) = 10 \times 0.2 + 20 \times 0.3 + 30 \times 0.5 = 23 $

NOTICE: Explanation for \( \mu = E(X) \)

In fact, the mean here is not different than the mean in statistics.

Consider the following ten numbers:
$ 10, 10, 20, 20, 20, 30, 30, 30, 30, 30 $

The probabilities to select 10, 20, or 30 are as in the table above.

The mean in statistics is also:
$ \mu = \frac{10 \times 2 + 20 \times 3 + 30 \times 5}{10} = 10 \times \frac{2}{10} + 20 \times \frac{3}{10} + 30 \times \frac{5}{10} = 23 $

◆ MEDIAN-MODE

These measures, known from statistics, are defined analogously:

MODE = The value \( X=a \) of the highest probability.
MEDIAN = The value \( X=m \) where the probability splits in two equal parts (0.5-0.5).

◆ VARIANCE (Only for HL )

We define:
$ \text{Var}(X) = E(X-\mu)^2 $
that is:
$ \text{Var}(X) = (x_1-\mu)^2 \times p_1 + (x_2-\mu)^2 \times p_2 + (x_3-\mu)^2 \times p_3 + \cdots $

An equivalent definition is:
$ \text{Var}(X) = E(X^2) – \mu^2 $
where:
$ E(X^2) = x_1^2 \times p_1 + x_2^2 \times p_2 + x_3^2 \times p_3 + \cdots $