Download Lecture Notes Chapter 12 rev04Nov14 (Moore)

Lecture Notes (Italics = Handouts)
Chapters 12 (Moore)
Binomial Distributions
Random variables were introduced in the text in chapter 9 (pages 176177). I didn’t need them to now so I saved them up.
Random Variables
A random variable assigns to each element of a sample space a
unique numerical value. We use uppercase italicized letters (e.g.
X, Y) to denote random variables and lowercase italicized letters
(e.g. x, y) to denote values of a random variable.
(3 coins, 3 M/C questions, two dice, randomly select a
student from the class)
discrete vs continuous (usually count vs measure)
(e.g. on students: ht, wt, # sibling, distance from home to
SCC, household income, # in household, ZIP code, units
completed)
Discrete probability distributions (can be given by a table,
formula, or a probability histogram)
For example: by table:
x
–3
–1
2
5
P(x)
0.2
0.4
0.3
0.1
by formula,
P(x) = x/10 for x = 1, 2, 3, 4
Properties: 0  P(x)  1 and ΣP(x) = 1
Mean and Standard Deviation of a discrete probability distribution
Mean and Variance of Discrete Random Variables (Moore)
Binomial Probability Distributions
Bernoulli experiment (exactly two outcomes, generically “success”
and “failure”, probability of success is denoted by p)
Binomial Model
Binomial experiment, n independent, identical Bernoulli trials
If X is the number of successes for a binomial experiment we then
X has a binomial probability distribution with n trial and the
probability of a success = p and we write X ~ Binom(n, p), (note
that the probability of a failure is denoted by q = 1 – p)
Probability for binomial distributions (Binomial Distributions)
n!
formula P(X = k) = nCk pk qn–k, where n Ck 
k !(n  k )!
table (rare now)
by calculator
For X ~ Binom(n, p) on the TI calculators
P(X = k) = binompdf(n,p,k)
P(X  k) = binomcdf(n,p,k)
P(X ≥ k) = 1 – binomcdf(n,p,k–1)
by computer (Maple, Excel, etc.)
Mean and standard deviation of a binomial distribution
 = np
 = npq
Note: for np  10 and nq  10 binomial approximately normal (see
Minitab Histograms)
Back in the mid-20th century (before modern calculators and computer
programs) it was common to use the fact the binomial distribution,
Binom(n, p) when np  10 and nq  10 can be closely
approximated by the Normal distribution N(np, npq ) to
approximate binomial probabilities. This is covered in the book on
pages 223 – 224, you can read it for historical purposes but you
won’t be tested on it.
Chapter 12 Exercises: 1 – 5, 7, 9, 12 – 19, 24 plus the handout “Binomial
Distributions”
Go to Chapter 18.