Discrete Uniform Distribution in Statistics

Discrete Uniform Distribution

By: Bindeshwar Singh Kushwaha
PostNetwork Academy

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Discrete Uniform Distribution

• A random variable \( X \) is said to have a discrete uniform distribution if it takes integer values from \( a \) to \( b \) with equal probability.
• The number of possible values is
  \[ n = b – a + 1. \]
• The probability mass function (pmf) is given by
  \[
  P(X = x) =
  \begin{cases}
  \dfrac{1}{n}, & x = a, a+1, \dots, b, \\
  0, & \text{otherwise.}
  \end{cases}
  \]
• Example: If \( X \) is the outcome of rolling an unbiased die, then \( a=1, b=6 \implies n=6 \).
• Hence,
  \[ P(X = x) = \frac{1}{6}, \quad x = 1,2,3,4,5,6. \]

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Mean of Discrete Uniform Distribution

• Let \( X \) be a discrete uniform random variable taking integer values from \( a \) to \( b \).
• The number of possible values is \( n = b – a + 1. \)
• The pmf is \( P(X = x) = \frac{1}{n}, \quad x = a, a+1, \dots, b. \)
• The mean is defined as
  \[ E[X] = \sum_{x=a}^{b} x \cdot \frac{1}{n}. \]
• Simplifying, we get
  \[ E[X] = \frac{1}{n} \sum_{x=a}^{b} x = \frac{a+b}{2}. \]

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Variance of Discrete Uniform Distribution

• The variance is
  \[ Var(X) = E[X^2] – (E[X])^2. \]
• Compute
  \[ E[X^2] = \frac{1}{n} \sum_{x=a}^{b} x^2. \]
• Using the formula for sum of squares,
  \[ E[X^2] = \frac{(b-a+1)(a^2 + ab + b^2)}{3n}. \]
• Hence,
  \[ Var(X) = \frac{(n^2 – 1)}{12}. \]

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Example 1: Mean and Variance of a Die Roll

• Let \( X \) be the number on an unbiased die. Then \( n = 6. \)
• Values: \( 1,2,3,4,5,6. \)
• \( P(X=x) = \frac{1}{6}. \)
• Mean: \( E[X] = \frac{n+1}{2} = \frac{7}{2}. \)
• Variance: \( Var(X) = \frac{n^2-1}{12} = \frac{35}{12}. \)

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Expected Frequency of Die Outcomes

Example 2: If an unbiased die is thrown 120 times, find the expected frequency of 1–6.

• \( P(X = x) = \dfrac{1}{6}. \)
• \( f(x) = N \cdot P(X=x) = 120 \times \dfrac{1}{6} = 20. \)
• Hence, each face is expected to appear 20 times.

X	P(X=x)	Expected Frequency f(x)
1	\(\frac{1}{6}\)	20
2	\(\frac{1}{6}\)	20
3	\(\frac{1}{6}\)	20
4	\(\frac{1}{6}\)	20
5	\(\frac{1}{6}\)	20
6	\(\frac{1}{6}\)	20

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Discrete Uniform Random Variable (1–10 Tickets)

• \( X \) is the number on a ticket from 1 to 10.
• \( P(X=x) = \frac{1}{10}, \ x = 1,2,\dots,10. \)
• \( N = 150. \)
• \( f(x) = N \cdot P(X=x) = 150 \cdot \frac{1}{10} = 15. \)

X	P(X = x)	Expected/Theoretical Frequency
1	\(\frac{1}{10}\)	15
2	\(\frac{1}{10}\)	15
3	\(\frac{1}{10}\)	15
4	\(\frac{1}{10}\)	15
5	\(\frac{1}{10}\)	15
6	\(\frac{1}{10}\)	15
7	\(\frac{1}{10}\)	15
8	\(\frac{1}{10}\)	15
9	\(\frac{1}{10}\)	15
10	\(\frac{1}{10}\)	15

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