Chapter 6.3: Binomial & Geometric Distributions - Summary
BINOMIAL DISTRIBUTIONS

Four Conditions (BINS)




2 outcomes – success and failure
Observations are independent.
We are counting the number of successes in a FIXED number of trials.
The probability of success, p, is the same for each observation

Notation: X is B(n, p) where n is the number of trials and p is the probability of success;
X = # of successes

Possible values of X (number of successes) are whole numbers from 0 to n.

n
P( x  k )    p k (1  p)n k
k 
Binomial formula:
(Given on exam)
n
n!
Where   =
or nCr (on calculator, where r = k)
 k  k!(n  k )!

Calculator commands:
 P(x = k) = binompdf(n,p,k)
 P(x  k) = binomcdf(n,p,k)
x  np

Expected # successes (Mean): E(x) =

Standard deviation:

Normal approximation of binomial distribution
(Given on exam)
 x  np(1  p) (Given on exam)
 When n is sufficiently large so that np  10 and n(1  p )  10 , a binomial
distribution is approximately NORMAL with a mean
x  np
and  x  np(1  p)
 If np  10 or n(1  p )  10 is not true, an exact binomial calculation should be used.

Construct probability distribution tables and histograms (for pdf and cdf)
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GEOMETRIC DISTRIBUTIONS

Conditions (1st 3 are the same as binomial)
 2 outcomes – success and failure
 Observations are independent.
 We are counting the number of trials needed to reach the FIRST success—NOT FIXED #
 The probability of success, p, is the same for each observation.

Possible values (number of trials to get the first success) are whole numbers from 1 to “infinity.”

Geometric formulas: (NOT on formula sheet)
 P(x = n) = (1 – p)n–1 • p (Probability that the 1st success occurs on the nth trial)
 P(x > n) = (1 – p)n


(Probability that it takes more than n trials to reach 1st success)
Calculator commands:
 P(x = k) = geometpdf(p,k)
 P(x  k) = geometcdf(p,k)
Expected # trials to reach first success (Mean): E(x) =
x 
1
p
Chapter 7: Sampling Distributions - Summary

Parameter – names a characteristic of the population.
Statistic – names a characteristic of a sample.
Parameters:
p = population proportion
Statistics:
µ = population mean

p̂ = sample proportion
x = sample mean
Sampling distribution of a statistic – the distribution of values taken by the statistic in all possible
samples of the same size from the same population.
 This is a theoretical distribution since it usually unfeasible to take the actual number of
samples needed to create this.

Unbiased – a statistic used to estimate a parameter is unbiased if the mean of its sampling
distribution is equal to the true value of the parameter being estimated.


 x is an unbiased estimate of

 pˆ is an unbiased estimate of p
 This does NOT mean that every sample mean, x , is equal to
or that every sample proportion,

p̂ is equal to p
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
Variability in the sampling distribution is reduced by increasing the sample size. As long as the
population is much larger than the sample (10 times as large), the spread of the sampling
distribution is approximately the same for any POPULATION size.
 Let’s say that again, the spread of the sampling distribution depends on the SAMPLE
size, not the POPULATION size.

Sampling distribution of sample proportions
 p̂  :
 Recognize when a problem involves a sample proportion
p̂ = successes = x
samplesize n
p̂
 When np  10 and n(1  p )  10 , the distribution of sample proportions is
approximately normal with
 p̂  p
and
 pˆ 
p(1  p)
n
(Given on the exam)
where p is the population proportion and n is the sample size
 The standard deviation formula is valid if the pop’n is at least 10 times the sample size.

Sampling distribution of sample means
x:
 If the sample size is large ( n  30 ) or the original population is normal, then the sampling
distribution of x is approximately Normal with
x  
and
where  is the population mean and
x 


(Given on exam)
n
is the population standard deviation.
 The standard deviation formula is valid if the population is at least 10 times the sample
size.


Sample size – both
 p̂ and  x are proportional to
1
n
 To divide
 p̂
or
x
 To divide
 p̂
or
 x by 3, the sample size must be increased by a factor of 9.
by 2 , the sample size must be increased by a factor of 4.
CENTRAL LIMIT THEOREM: Regardless of the shape of the original population
distribution, the distribution of the sample means x is approximately Normal when the sample
size is large ( n  30 ).
 This allows us to use Normal calculations for x
N(µ,

) even when the original
x
population is not Normally distributed, as long as n is large enough!
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