Transcript Ch 8 Notes
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
nk
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