Download Binomial Distribution

Binomial Distribution
The binomial distribution
discrete distribution.
is
a
Binomial Experiment
 A binomial experiment has the following properties:

experiment consists of n identical and independent trials

each trial results in one of two outcomes: success or failure
 P(success) = p
 P(failure) = q = 1 - p for all trials

The random variable of interest, X, is the number of successes
in the n trials.

X has a binomial distribution with parameters n and p
EXAMPLES
 A coin is flipped 10 times. Success = head.
 X =
n=
p=
 Twelve pregnant women selected at random, take a home
pregnancy test. Success = test says pregnant.
 X =
n=
p=?
 Random guessing on a multiple choice exam. 25 questions.
4 answers per question. Success = right answer.
 X =
n=
p=
Examples when assumptions do not hold
 Basketball player shoots ten free throws

Feedback affects independence and constant p
 Barrel contains 3 red apples and 4 green apples;
select 4 apples without replacement; X = # of red
apples.

Without replacement implies dependence
What is P(x) for binomial?
n!
x n− x
p q
P( x) =
x!(n − x)!
Mean and Standard Deviation
 The mean (expected value) of a binomial
random variable is µ = np
 The standard deviation of a binomial random
variable is σ = npq
Example
 Random Guessing; n = 100 questions.



Probability of correct guess; p = 1/4
Probability of wrong guess; q = 3/4
1
Expected Value = µ= np= 100  =
 25
4


 On average, you will get 25 right.
 Standard Deviation =
=
σ
=
npq
− p)
np (1=
 1  3 
100   =
 4.33
 4  4 
Example
 Cancer Treatment; n = 20 patients
 Probability of successful treatments; p = 0.7
 Probability of no success; q = ?
 Calculate the mean and standard deviation.
Normal Approximation
 For large n, the binomial distribution
can be approximated by the normal,
X − np
Z=
npq
is approximately standard normal for
large n.