Interactive Concept Exercise — Exponential

Paste data → histogram → MLE fit \( \hat\lambda=1/\bar{x} \) → CDF & probabilities → Excel.

Theme:

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) \(P(X\lt 5)\)

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)