Edexcel IAL - Statistics 1- 5.4 Discrete Uniform Distribution- Study notes - New syllabus
Edexcel IAL – Statistics 1- 5.4 Discrete Uniform Distribution-Study notes- New syllabus
Edexcel IAL – Statistics 1- 5.4 Discrete Uniform Distribution-Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 5.4 Discrete Uniform Distribution
The Discrete Uniform Distribution
A discrete uniform distribution is a discrete probability distribution in which all possible values of the random variable are equally likely.
This distribution is commonly used to model situations such as fair dice, fair spinners, or any experiment where each outcome has the same probability.
Definition
A discrete random variable \( X \) is said to have a discrete uniform distribution on the integers
\( a, a+1, a+2, \dots , b \)
if
\( P(X = x) = \dfrac{1}{b – a + 1}, \quad x = a, a+1, \dots , b \)
All other values of \( X \) have probability zero.
Mean of the Discrete Uniform Distribution
For a discrete uniform distribution on
\( a, a+1, \dots , b \)
the mean (expected value) is
\( E(X) = \dfrac{a + b}{2} \)
This is simply the midpoint of the smallest and largest values.
Variance of the Discrete Uniform Distribution
The variance of a discrete uniform distribution on
\( a, a+1, \dots , b \)
is given by
\( \mathrm{Var}(X) = \dfrac{(b – a + 1)^2 – 1}{12} \)
These results may be used directly without proof.
Key Features
- All outcomes have equal probability
- The mean lies exactly halfway between the extreme values
- The spread depends on the number of possible values
Example :
A fair six-sided die is rolled once. Let \( X \) be the number shown.
State the distribution of \( X \), and find \( E(X) \) and \( \mathrm{Var}(X) \).
▶️ Answer/Explanation
\( X \) has a discrete uniform distribution on \( 1,2,3,4,5,6 \).
\( E(X) = \dfrac{1 + 6}{2} = 3.5 \)
\( \mathrm{Var}(X) = \dfrac{6^2 – 1}{12} = \dfrac{35}{12} \)
Conclusion: The mean is 3.5 and the variance is \( \dfrac{35}{12} \).
Example :
A random variable \( X \) is uniformly distributed on the integers \( 4,5,6,7,8 \).
Find \( E(X) \) and \( \mathrm{Var}(X) \).
▶️ Answer/Explanation
\( a = 4,\; b = 8 \)
\( E(X) = \dfrac{4 + 8}{2} = 6 \)
\( \mathrm{Var}(X) = \dfrac{5^2 – 1}{12} = 2 \)
Conclusion: The mean is 6 and the variance is 2.
Example :
A discrete random variable \( X \) has a uniform distribution on the integers \( 1 \) to \( n \). Given that \( E(X) = 6 \), find the value of \( n \).
▶️ Answer/Explanation
Using the mean formula:
\( \dfrac{1 + n}{2} = 6 \)
\( n = 11 \)
Conclusion: The value of \( n \) is 11.