Lesson 5: The Uniform Distribution
Understanding distributions where all outcomes are equally likely
Introduction to the Uniform Distribution
While the normal distribution is characterized by values clustering around the mean, the uniform distribution represents the opposite scenario: a situation where all outcomes are equally likely.
Definition: Uniform Distribution
A continuous uniform distribution, denoted as U(a, b), is a probability distribution where:
- All values between minimum (a) and maximum (b) are equally likely
- The probability density is constant across the entire range
- The graph is a rectangle (flat, not curved)
- Values outside the range [a, b] have probability zero
The uniform distribution is sometimes called the rectangular distribution because of its flat, rectangular shape when graphed.
- Normal distribution: Bell-shaped curve with values clustering at the center
- Uniform distribution: Flat rectangle with all values equally likely
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Imagine a flat horizontal line (rectangle)
The height is constant from a to b, creating a rectangular shape.
Properties of the Uniform Distribution
The continuous uniform distribution U(a, b) is completely determined by two parameters:
- a = minimum value (left endpoint)
- b = maximum value (right endpoint)
Probability Density Function (PDF)
f(x) = 1 / (b − a)
for a ≤ x ≤ b
(and f(x) = 0 for x outside this range)
This constant height of 1/(b−a) ensures that the total area under the curve equals 1 (since area = base × height = (b−a) × 1/(b−a) = 1).
Mean (Expected Value)
μ = (a + b) / 2
The mean is the midpoint between a and b
This makes intuitive sense: if all values are equally likely, the average should be right in the middle.
Variance and Standard Deviation
σ² = (b − a)² / 12
Variance
σ = (b − a) / √12 ≈ (b − a) / 3.464
Standard Deviation
Calculating Probabilities
Because the uniform distribution is rectangular, calculating probabilities is straightforward—it's simply finding the area of a rectangle.
P(x₁ < X < x₂) = (x₂ − x₁) / (b − a)
Area = base × height = (x₂ − x₁) × [1/(b−a)]
- Base: The length of the interval (x₂ − x₁)
- Height: The constant probability density 1/(b−a)
- Probability: Base × Height = the area of the rectangle
Special cases:
- P(X < x₁) = (x₁ − a) / (b − a)
- P(X > x₂) = (b − x₂) / (b − a)
- P(X = any specific value) = 0 (continuous distribution)
Examples
Example 1: Bus Wait Time
Scenario: A bus arrives every 15 minutes. You arrive at the bus stop at a random time. Your wait time X is uniformly distributed: X ~ U(0, 15) minutes.
(a) What is the probability you wait between 5 and 10 minutes?
Solution:
- a = 0, b = 15
- P(5 < X < 10) = (10 − 5) / (15 − 0) = 5 / 15 = 1/3 ≈ 0.333
- Answer: About 33.3%
(b) What is the probability you wait less than 3 minutes?
Solution:
- P(X < 3) = (3 − 0) / (15 − 0) = 3 / 15 = 1/5 = 0.20
- Answer: 20%
(c) What is the average wait time?
Solution:
- μ = (a + b) / 2 = (0 + 15) / 2 = 7.5 minutes
- Answer: 7.5 minutes
Example 2: Random Number Generator
Scenario: A computer generates random numbers uniformly distributed between 0 and 1: X ~ U(0, 1)
(a) What is the probability the number is greater than 0.75?
Solution:
- P(X > 0.75) = (1 − 0.75) / (1 − 0) = 0.25 / 1 = 0.25
- Answer: 25%
(b) What is the standard deviation?
Solution:
- σ = (b − a) / √12 = (1 − 0) / √12 = 1 / 3.464 ≈ 0.289
- Answer: σ ≈ 0.289
(c) What is the probability the number is exactly 0.5?
Solution:
- For continuous distributions, P(X = any specific value) = 0
- Answer: 0 (probability zero)
Example 3: Manufacturing Tolerance
Scenario: A machine cuts metal rods to a target length of 100 cm. Due to random variation, the actual length is uniformly distributed between 99.5 and 100.5 cm: X ~ U(99.5, 100.5)
(a) What percentage of rods are between 99.8 and 100.2 cm?
Solution:
- a = 99.5, b = 100.5
- P(99.8 < X < 100.2) = (100.2 − 99.8) / (100.5 − 99.5) = 0.4 / 1.0 = 0.40
- Answer: 40%
(b) What is the mean length?
Solution:
- μ = (99.5 + 100.5) / 2 = 200 / 2 = 100 cm
- Answer: 100 cm (exactly on target)
(c) What is the variance?
Solution:
- σ² = (b − a)² / 12 = (100.5 − 99.5)² / 12 = 1² / 12 = 1/12 ≈ 0.0833 cm²
- Answer: σ² ≈ 0.0833 cm²
Example 4: Comparing Uniform to Normal
Scenario: Two processes produce items with mean 50 and range of 40-60:
- Process A: Uniform distribution U(40, 60)
- Process B: Normal distribution N(50, σ) where we set σ so 40 and 60 are ±3σ from mean
Question: Which process produces more values between 45 and 55?
Solution:
Process A (Uniform):
- P(45 < X < 55) = (55 − 45) / (60 − 40) = 10 / 20 = 0.50
- 50% of values
Process B (Normal):
- If 60 = μ + 3σ, then 60 = 50 + 3σ, so σ ≈ 3.33
- 45 = 50 − 5 = μ − 1.5σ
- 55 = 50 + 5 = μ + 1.5σ
- By the Empirical Rule (interpolating), approximately 86.6% of values fall within 1.5 SD
Answer: Process B (normal) produces more values between 45 and 55 because values cluster around the mean. Process A (uniform) spreads values evenly across the entire range.
When to Use the Uniform Distribution
The uniform distribution is appropriate when:
Use Uniform Distribution When:
- All outcomes are equally likely (no value is more probable than any other)
- You have a bounded range with known minimum and maximum values
- There's no reason to expect clustering around any particular value
- The process is truly random within the specified bounds
Common examples:
- Wait times: Arriving randomly during a fixed time interval
- Random number generation: Computer-generated random values
- Round-off errors: Measurement rounded to nearest unit
- Particle position: Object equally likely anywhere in a region
- Manufacturing: Random variation within tight tolerances (when well-controlled)
Uniform vs Normal Distribution
Understanding when to use each distribution is crucial:
| Characteristic | Uniform Distribution | Normal Distribution |
|---|---|---|
| Shape | Rectangular (flat) | Bell-shaped (curved) |
| Probability | All values equally likely | Values near mean more likely |
| Parameters | a (min), b (max) | μ (mean), σ (standard deviation) |
| Range | Finite: [a, b] | Infinite: (−∞, +∞) |
| Standard Deviation | Determined by range: (b−a)/√12 | Independent parameter σ |
| Symmetry | Symmetric around midpoint | Symmetric around mean |
| When to Use | Truly random within bounds, no clustering expected | Natural variation, values cluster around mean |
Real-World Application Guide
Use Uniform:
- Spinner landing on any position 0° to 360°
- Arrival time during a 1-hour window with no pattern
- Random selection from a bounded interval
Use Normal:
- Human heights, weights, IQ scores
- Measurement errors in scientific experiments
- Test scores, blood pressure readings
Check Your Understanding
Try these questions to test what you've learned in this lesson.
Question 1: A subway train arrives every 10 minutes. If you arrive at a random time, what is the probability you wait more than 7 minutes? (Assume wait time is uniform.)
Answer: 0.30 or 30%
Explanation: X ~ U(0, 10). P(X > 7) = (10 − 7) / (10 − 0) = 3/10 = 0.30
Question 2: For a uniform distribution U(20, 50), what is the mean?
Answer: 35
Explanation: μ = (a + b) / 2 = (20 + 50) / 2 = 70 / 2 = 35. The mean is always the midpoint.
Question 3: True or False: In a uniform distribution, the probability of getting a value between 10 and 20 is the same as getting a value between 30 and 40 (assuming both intervals are within the range).
Answer: True
Explanation: Since all values are equally likely in a uniform distribution, any two intervals of the same length have the same probability. Both intervals have length 10, so P(10 < X < 20) = P(30 < X < 40).
Question 4: A random number generator produces values uniformly between 0 and 100. What is the standard deviation?
Answer: σ ≈ 28.87
Explanation: σ = (b − a) / √12 = (100 − 0) / √12 = 100 / 3.464 ≈ 28.87
Question 5: Which statement best describes the key difference between uniform and normal distributions?
Answer: Uniform has all values equally likely (flat graph), while normal has values clustering around the mean (bell-shaped graph).
Explanation: This is the fundamental distinction. Uniform is rectangular with constant probability density, while normal is bell-shaped with higher probability near the center.
Key Takeaways from Lesson 5
- Uniform distribution U(a, b) means all values between a and b are equally likely
- Graph is rectangular (flat), not bell-shaped
- Probability density: f(x) = 1/(b−a) (constant)
- Mean is the midpoint: μ = (a+b)/2
- Standard deviation: σ = (b−a)/√12 (determined by range)
- Probability is calculated as area of rectangle: (length of interval) / (total range)
- Use uniform when all outcomes are truly equally likely within known bounds
- Use normal when values cluster around a mean (most natural phenomena)