📘 Session 4.4 – Continuous Uniform Distribution

🔧 Why uniform? Two quick, real-life connections

Engineering — tolerance window (drilled hole)

A CNC operation is set to drill a hole and auto-reject if it’s outside 12.50 ± 0.10 mm. With process control keeping outcomes spread fairly evenly in the allowed window, a quick first model is X ~ Uniform(12.40, 12.60). Then any sub-interval’s probability is just its length divided by 0.20.

Student life — random quiz question number

Your instructor lets a script randomly pick the starting question number from 1 to 10 uniformly. The chance it starts between 3 and 6 is length 4 over total length 10 → 4/10 = 0.4.

Definition: A continuous random variable X has a Uniform Distribution on the interval [a, b] if its probability density function (PDF) is:
f(x) = 1 / (b − a), for a ≤ x ≤ b

📏 Properties

Mean: E(X) = (a + b) / 2
Variance: Var(X) = (b − a)² / 12

📊 Example 4.7 – Uniform Current

Context: The current X in a wire is uniformly distributed between 4.9 and 5.1 milliamperes. The PDF is:
f(x) = 5 for 4.9 ≤ x ≤ 5.1
Question: What is the probability that X is between 4.95 and 5.0?
Solution:
P(4.95 < X < 5.0) = ∫_4.95^5.0 5 dx = 5 × (5.0 − 4.95) = 0.25

🧪 Try it yourself (theory vs simulation)

a (left endpoint) b (right endpoint) L (interval start) U (interval end) Samples (for histogram)

Theoretical:
Simulated:

Theory for Uniform is simply length of interval divided by total length: (U–L)/(b–a), clipped to [0,1].

📈 Cumulative Distribution Function (CDF)

The cumulative distribution function F(x) is:

F(x) = 0 for x < a
F(x) = (x − a) / (b − a) for a ≤ x < b
F(x) = 1 for x ≥ b

📊 Excel: sample, estimate probability, and make a histogram

Generate Uniform(a,b) samples
Option A (constants, here a=4.9, b=5.1). Put in A2 and fill down:
```
=4.9 + (5.1 - 4.9) * RAND()
```
Option B (use cells for a and b, e.g. A1=a, B1=b):
```
=$A$1 + ($B$1 - $A$1) * RAND()
```
Tip: To “freeze” values, copy the column and Paste Special → Values.
Estimate P(L ≤ X ≤ U)
With samples in A2:A5001 and bounds in cells L and U:
```
=COUNTIFS(A2:A5001, ">="&L, A2:A5001, "<="&U) / ROWS(A2:A5001)
```
Use cell references for L and U (e.g., C1, D1).
Mean and variance from data
```
=AVERAGE(A2:A5001)
```
```
=VAR.P(A2:A5001)
```
For Uniform(a,b), the true variance is $(b-a)^2/12$ — your estimate should be close.
Histogram
Excel 365: Insert → Chart → Histogram.
Or build bins and use COUNTIFS, then insert a column chart.

🧠 Summary

The continuous uniform distribution models a flat, even likelihood over an interval.
The PDF is constant and the total area under the curve equals 1.
Use this model when every outcome in an interval is equally likely.