Uniform Distribution (Edexcel A-Level Further Mathematics): Revision Notes

Example: A dice is thrown, and the number on the dice is noted. The following table details the probability distribution of this, where $X$ is "the score obtained":

$\begin{array}{c|c|c|c|c|c|c} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P(X = x) & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} \end{array}$

This is an example of a uniform distribution and can be more succinctly written as: $X \sim U(6)$

Example: Put $X \sim U(2)$ in a table.

Example : A fair spinner has outcomes 2, 4, 6, or 8. Find the expected mean score and expected variance after a large number of spins.

(Note: This situation is not quite a uniform distribution because the outcomes are not consecutive integers starting at $1$. However, for now, let's pretend they are.)

$X \sim U(4)\ implies:$

$E(X) = \frac{4 + 1}{2} = :success[2.5]$

$\text{Var}(X) = \frac{1}{12} (4^2 - 1) = :success[1.25]$

However, this is not the mean and variance we need. If we consider $E(2X)$ and $\text{Var}(2X)$ , we do get the quantities we need since the outcomes are double that of $X \sim U(4)$ .

$E(2X) = 2E(X) = :success[5]$

$\text{Var}(2X) = 2^2 \text{Var}(X) = :success[5]$

Example: A fair die is numbered 7, 10, 13, 16, 19, 22. Find the expected score per throw and the variance after a large number of throws.

$X \sim U(6)\ implies:$

$E(X) = \frac{1 + 6}{2} = :success[\frac{7}{2}]$

$\text{Var}(X) = \frac{1}{12} (6^2 - 1) = :success[\frac{35}{12}]$

If $X \sim U(6)$ , then the outcomes are described by $3X + 4$ .

$E(3X + 4) = 3E(X) + 4 = 3 \times \frac{7}{2} + 4 = :success[\frac{29}{2}]$

$\text{Var}(3X + 4) = 3^2 \text{Var}(X) = 9 \times \frac{35}{12} = :success[\frac{105}{4}]$