📘 Session 3.4 - Discrete Uniform Distribution

📐 Definition

A random variable X has a discrete uniform distribution if it can take on a finite number of equally likely values.

Let X take values {a, a+1, a+2, ..., b}, where a and b are integers with a ≤ b. Then:

• Number of possible values = n = b − a + 1
• Probability of each outcome: f(x) = 1 / n for x ∈ {a, ..., b}

📊 Deriving the Mean

By definition: E(X) = Σ x·f(x)
Since each f(x) = 1/n, we have:

E(X) = (1/n) · Σ from x=a to b of x
     = (1/n) · [ (a + b)(n) / 2 ]
     = (a + b) / 2
    

✔️ The mean is simply the midpoint of a and b.

📊 Deriving the Variance

First, compute E(X²):

E(X²) = (1/n) · Σ from x=a to b of x²
    

Using the formula for the sum of squares: Σ_{k=1}^m k² = m(m+1)(2m+1)/6, we can express E(X²) in closed form.

After simplification, the variance of a discrete uniform distribution is:

Var(X) = [(n² − 1)] / 12
where n = (b − a + 1)
    

📊 Standard Deviation

σ = √Var(X) = √( (n² − 1) / 12 )
    

✅ Summary of Formulas

• Mean: μ = (a + b) / 2
• Variance: σ² = ( (b − a + 1)² − 1 ) / 12
• Standard Deviation: σ = √[ ( (b − a + 1)² − 1 ) / 12 ]

🎲 Example 3.10 – Serial Number

The first digit of a part's serial number is equally likely to be any digit from 0 to 9.

This gives a uniform distribution with 10 values, each with probability 0.1.

📞 Example 3.11 – Number of Voice Lines

Let X be the number of voice lines in use at a given time. X is uniformly distributed over {0, 1, ..., 48}.

• 📌 E(X) = (0 + 48) / 2 = 24
• 📌 Var(X) = [(49² − 1)/12] = 2400/12 = 200
• 📌 σ = √200 ≈ 14.14

Interpretation: On average, 24 voice lines are in use. However, the high standard deviation means that usage can vary widely, sometimes far above or below the average.

📊 Example 3.12 – Proportion of Voice Lines

Let X be the number of voice lines used out of 48, and define a new variable Y = X / 48, the proportion of lines used. Since X is uniformly distributed over {0, 1, ..., 48}, we can use linear transformation formulas:

• E(Y) = (1/48) · E(X) = (1/48) · 24 = 0.5
• Var(Y) = (1/48)² · Var(X) = (1/2304) · 200 = 0.0868
• σ(Y) = √0.0868 ≈ 0.295

Explanation: Expectation is linear, so scaling X by 1/48 scales the mean by 1/48. Variance scales by the square of the factor: Var(aX) = a²Var(X).

✅ Even though the average proportion is 50%, the standard deviation tells us that actual usage might often fall between roughly 20.5% and 79.5%.

📌 This kind of transformation is important when comparing different systems with different total capacities using normalized units like percentages or proportions.

🧠 Practice Question

X ~ Uniform(1, 7). Find:

1. Mean μ
2. Variance σ²
3. Standard Deviation σ

Show Answer