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)