Chapter 8
The Binomial and
Geometric Distributions
YMS 8.1
The Binomial Distributions
Binomial Distribution
 Distribution of the count X of successes
in the binomial setting with parameters n
and p
 n is the number of observations and p is the
probability of a success on any one
observation
 The possible values of X are the whole
numbers from 0 to n
 Denoted by B(n,p).
Binomial Setting
 Each observation falls into one of just two
categories, “success” or “failure”
 There is a fixed number n of observations
 The n observations are all independent
 The probability of success, p, is the same
for each observation
Binomial Calculations
 pdf
 probability distribution function which
assigns value to single outcome X
 cdf
 cumulative distribution function which
assigns value to range of X
 Be careful of calculator entries when
finding greater than or at least!
p445 #8.4-8.8
Vocab and Simulations
 Combination
 order doesn’t matter
 n choose k
 Factorial
 n! = n x (n-1) x (n-2) x 3 x 2 x 1 and 0! = 1
 Use randbin(1, p, n) to give 1 p% of the
time and 0 (1-p)% of the time
Using Binomials
 Binomial Probability
n
k
nk
P
(
X

k
)

(
)
p
(1

p
)

where the
k
coefficient is a combination
 Binomial Mean & Standard Deviation
  np
  np(1  p)
Normal approximation to
binomial distributions
 As the number of trials n gets larger, the
binomial distribution gets close to a
normal distribution
 Rule of thumb
 N(np, np(1 p) ) can be used when n and p
satisfy np ≥ 10 and n(1 – p) ≥ 10
p454 #8.16-8.19
HW: 8.12-8.14, 8.26, 8.32-8.36
YMS 8.2
The Geometric Distribution
Geometric Setting
 Each observation falls into one of just two
categories: “success” or “failure”
 The probability of success, p, is the same
for each observation
 The n observations are all independent
 The variable of interest is the number of
trials required to obtain the first
success
Calculating geometric
probabilities
 probability that the first success occurs
on the nth trial is
P( X  n)  (1  p)
n 1
p
 probability that it takes more than n trials
to see the first success is
P( X  n)  (1  p)
n
Geometric Mean &
Standard Deviation
1

p

1 p
2
p
8.37 and 8.40
HW: p474 #8.44-8.46
Review: p479 #8.55-8.56, 8.60, 8.62-8.63