AP Statistics

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Binomial and Geometric Distributions

Chapter 8

Definitions of a Binomial and Geometric Distributions

• Definition of a Binomial Distribution

– The distribution of the count X of successes in the binomial setting is the Binomial Distribution.

– X is B(n,p)

• n is the number of observations.

• P is the probability of each of the observations

• Definition of a Geometric Distribution

– The distribution of the count n of trials in the geometric setting is the Geometric Distribution.

– X is G(n,p)

• n is the number of observations.

• P is the probability of how long the success would take to happen once.

Definitions of Settings

• The Binomial Setting

– Each observation falls into one of just two categories, which for convenience are called “successes” and “failures”

– There is a fixed number of n observations

– All observations are independent.

– The probability is the same for each observation.

• The Geometric Setting– Each observation falls into

one of just two categories, which for convenience are called “successes” and “failures”

– All observations are independent.

– The probability is the same for each observation.

– The variable of interest is the number of trials required to

obtain the first success.

Binomial Distributions

• A Binomial Distribution has a probability distribution function…or a Binomial PDF.

• The Binomial PDF is located on a TI calculator under 2nd / DISTR / 0:binompdf.

• This is used to find the binomial probability of a value X. So, it is used to find an exact match to a number X.

• A Binomial Distribution has a cumulative distribution function… or a Binomial CDF.

• The Binomial CDF is located on a TI calculator under 2nd / DISTR / A:binomialcdf.

• This is used to find the binomial probability of a value X. So, it is used to find the added probabilities up to a number X.

Binomial and Geometric PDF and CDF

• Binomial PDF is written as binomialpdf(n, p, x)

– n is trials

– p is probability

– x is number you are looking for

• Binomial CDF is written as binomialcdf(n, p, x)

– n is trials

– p is probability

– x is number you looking for up to that number

• Geometric PDF is written as geometricpdf(n, p, x)

– n is number of trials required to reach one success

– p is probability– x is number of successes needed

• Geometric CDF is written as geometriccdf(n, p, x)

– n is number of trials required to reach one success

– p is probability– x is number of successes

needed

Important Notes

• The approximation is based on finding an area under the normal curve that approximates the y-coordinate of the appropriate binomial probability plot.

• In order to use the normal approximation to answer binomial questions the binomial is discrete and the normal is continuous, the values of interest must sometimes be adjusted.

• When n is large, it is unnecessary to apply the continuity correction for an interval of values since the error introduced by not applying it is very small.

A Few Shortcuts and Hints1-seq (x, x, 0, 10, 1) L1

2-binompdf (10, .1, L1) L2 will calculate each P(x =…)

L1 L2 L3

1. .348682. .387423. .193714. .05745. …6. …

3- Turn off any Y= “stuff”

4-[STAT] [PLOT]… histogramX list = L1 Y list = L2

5- Set window X[-.5, 10.5] Y[-.1, .45]