📊 Chapter 3 Review – Discrete Random Variables & Distributions
  🔑 Core Ideas
  
  
Excel with a PMF table:
Assume x-values in A2:A10 and probabilities in B2:B10 (with SUM(B2:B10)=1)
Mean (E[X])             =SUMPRODUCT(A2:A10, B2:B10)
Second moment E[X^2]     =SUMPRODUCT((A2:A10)^2, B2:B10)
Variance                 =SUMPRODUCT((A2:A10)^2,B2:B10) - (SUMPRODUCT(A2:A10,B2:B10))^2
Std dev                  =SQRT( SUMPRODUCT((A2:A10)^2,B2:B10) - (SUMPRODUCT(A2:A10,B2:B10))^2 )
Check Σp(x)=1            =SUM(B2:B10)
  
 
  🎲 Discrete Uniform Distribution
  Model: X takes each integer from a to b (inclusive) with equal probability.
  f(x) = 1 / (b − a + 1),  x = a, a+1, …, b
  E[X] = (a + b)/2
  Var(X) = { (b − a + 1)² − 1 } / 12
  
Mean     =(a+b)/2
Variance =(((b-a+1)^2)-1)/12
PMF for x (if you list the support in A2:A? and set probs in B2:B?): set all B cells to =1/(b-a+1)
  
 
  🧪 Binomial Distribution — X ~ Binomial(n, p)
  Model: Number of successes in n independent Bernoulli trials (success prob. p).
  P(X = x) = C(n, x) p^x (1 − p)^{n − x}, x = 0,1,…,n
  E[X] = n p, Var(X) = n p (1 − p)
  
PMF (exactly x)         =BINOM.DIST(x, n, p, FALSE)
CDF (≤ x)               =BINOM.DIST(x, n, p, TRUE)
Right tail (≥ k)        =1 - BINOM.DIST(k-1, n, p, TRUE)
Range a..b              =BINOM.DIST(b,n,p,TRUE) - BINOM.DIST(a-1,n,p,TRUE)
Quantile (min x with F≥α)=BINOM.INV(n, p, α)
  
  Normal approx. if n large and p not extreme (use continuity correction). Poisson approx. if n large, p small, with λ=np.
 
  🎯 Geometric Distribution — “Trials until first success”
  Model: X = number of trials needed for the first success. Support: 1,2,3,…
  P(X = k) = (1 − p)^{k − 1} p
  F(k) = P(X ≤ k) = 1 − (1 − p)^k
  E[X] = 1/p, Var(X) = (1 − p) / p²
  
Excel (via Negative Binomial with r=1):
PMF at k        =NEGBINOM.DIST(k-1, 1, p, FALSE)
CDF at k (≤k)   =NEGBINOM.DIST(k-1, 1, p, TRUE)
  
  “Lack of memory”: P(X > s+t | X > s) = P(X > t).
 
  📦 Negative Binomial — X = trials to get r successes
  Model: Counts trials until the r-th success (r = 1 recovers geometric).
  P(X = k) = C(k−1, r−1) p^r (1 − p)^{k − r}, k = r, r+1, …
  E[X] = r/p, Var(X) = r(1 − p)/p²
  
Excel note: NEGBINOM.DIST uses “number of failures” (f) before r successes.
If X = total trials, then f = X − r.
PMF at X=k      =NEGBINOM.DIST(k - r, r, p, FALSE)
CDF at X=k      =NEGBINOM.DIST(k - r, r, p, TRUE)
  
 
  📈 Poisson Distribution — X ~ Poisson(λ)
  Model: Counts of events in a fixed interval when events occur independently at constant average rate λ.
  P(X = x) = e^{−λ} λ^x / x!, x = 0,1,2,…
  E[X] = λ, Var(X) = λ
  
PMF (exactly x)  =POISSON.DIST(x, lambda, FALSE)
CDF (≤ x)        =POISSON.DIST(x, lambda, TRUE)
Right tail (≥ k) =1 - POISSON.DIST(k-1, lambda, TRUE)
Range a..b       =POISSON.DIST(b,lambda,TRUE) - POISSON.DIST(a-1,lambda,TRUE)
  
  Binomial(n,p) ≈ Poisson(λ=np) when n is large and p is small (λ moderate).
 
  🧠 Quick Checks & Mini Examples
  1) PMF sanity check
  Given a table of x and f(x), verify f(x)≥0 and Σ f(x)=1. If not, it’s not a valid PMF.
  2) Discrete Uniform
  Roll a fair die (a=1, b=6): E[X]=(1+6)/2=3.5; Var(X)=((6)^2−1)/12=35/12≈2.917.
  3) Binomial
  20 T/F questions, random guessing: n=20, p=0.5 ⇒ E[X]=10, Var=5. P(X=10)=BINOM.DIST(10,20,0.5,FALSE)≈0.176.
  4) Geometric
  Guessing 1 of 4 choices: p=0.25 ⇒ E[X]=4. P(X=3)=(0.75)^2*0.25=0.140625.
  5) Negative Binomial
  Trials to 5th success, p=0.25: E[X]=20, Var=60. P(X=10)=NEGBINOM.DIST(10-5,5,0.25,FALSE)≈0.0146.
  6) Poisson
  ER arrivals rate λ=3/hour. P(X=5)=POISSON.DIST(5,3,FALSE)≈0.1008. P(X≤2)=POISSON.DIST(2,3,TRUE)≈0.423.
 
  📎 Excel Cheat Sheet (Chapter 3)
  
    
      Generic PMF table
      
E[X]     =SUMPRODUCT(A2:A10,B2:B10)
E[X^2]   =SUMPRODUCT((A2:A10)^2,B2:B10)
Var(X)   =SUMPRODUCT((A2:A10)^2,B2:B10) - (SUMPRODUCT(A2:A10,B2:B10))^2
σ        =SQRT( above )
Check    =SUM(B2:B10)
      
      Discrete Uniform
      
Mean     =(a+b)/2
Var      =(((b-a+1)^2)-1)/12
      
     
    
      Distribution functions
      
Binomial PMF      =BINOM.DIST(x, n, p, FALSE)
Binomial CDF      =BINOM.DIST(x, n, p, TRUE)
Binomial Quantile =BINOM.INV(n, p, α)
Geometric PMF     =NEGBINOM.DIST(k-1, 1, p, FALSE)
Geometric CDF     =NEGBINOM.DIST(k-1, 1, p, TRUE)
NegBin PMF (X=k)  =NEGBINOM.DIST(k - r, r, p, FALSE)
NegBin CDF        =NEGBINOM.DIST(k - r, r, p, TRUE)
Poisson PMF       =POISSON.DIST(x, lambda, FALSE)
Poisson CDF       =POISSON.DIST(x, lambda, TRUE)
Right tail (≥k)   =1 - POISSON.DIST(k-1, lambda, TRUE)
      
     
   
  Remember parameterizations: Excel’s NEGBINOM.DIST(number_f, trials, probability_s, cumulative) uses number of failures before r successes. If your variable is “total trials” X, then number_f = X − r.