1) Data (positive values)
Think inter-arrival times, wait times, lifetimes (no negatives). Example uses restaurant wait times.
n
—
mean \(\bar{x}\)
—
st.dev
—
λ̂ = — (mean ≈ —)
Model: \( f(x)=\lambda e^{-\lambda x},\; F(x)=1-e^{-\lambda x},\; x\ge0 \)
2) Histogram + PDF Overlay
Bars show density of your data; curve overlays the fitted PDF using \( \hat\lambda \).
3) Why Exponential (not Normal)?
- Support: Exponential supports \(x\ge0\). Normal allows negatives (nonsense for waiting times).
- Shape: Exponential is right-skewed (many short waits, few long). Normal is symmetric.
- Memoryless: \(P(X> s+t\mid X> s)=P(X> t)\)—unique to exponential; normal is not memoryless.
- Quick diagnostic: Exponential has \( \mathrm{sd}\approx \mathrm{mean} \) (since \( \mathrm{sd}=1/\lambda = \) mean).
Data min ≥ 0?
—
Skewness \(g_1\) > 0?
—
mean ≈ sd ?
—
Median < Mean?
—
If several checks fail, consider Lognormal or Gamma instead.
4) Math Work — How do we get \( \hat\lambda \)?
Step 1: \( \bar{x}=\frac{1}{n}\sum x_i \)
Step 2: For Exponential(\(\lambda\)),
\[
\ell(\lambda)=n\ln\lambda-\lambda \sum x_i
\Rightarrow \frac{d\ell}{d\lambda}=\frac{n}{\lambda}-\sum x_i=0
\Rightarrow \hat\lambda=\frac{n}{\sum x_i}=\frac{1}{\bar{x}}.
\]
From your data: \(\bar{x}=\) — ⇒ \( \hat\lambda=\) —
Interpretation: \(E[X]=1/\lambda\), \( \mathrm{Var}(X)=1/\lambda^2\).
Memoryless: remaining wait doesn’t depend on time already waited.
5) CDF and Four Core Probabilities
λ̂ = —
A) \(P(X\lt 5)\)
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—
—
B) \(P(3\lt X\lt 5)\)
—
—
—
C) \(P(X\gt 5)\)
—
—
—
D) \(P(X\gt 7)\)
—
—
—
6) Excel — Replicate the Results
λ: =1/AVERAGE(A2:A101)
P(X<5): =EXPON.DIST(5,λ,TRUE)
P(3<X<5): =EXPON.DIST(5,λ,TRUE)-EXPON.DIST(3,λ,TRUE)
P(X>5): =1-EXPON.DIST(5,λ,TRUE)
P(X>7): =1-EXPON.DIST(7,λ,TRUE)