Transcript Binomial and Geometric Distributions

AP Statistics
Powerpoint created, constructed, mocked, fabricated, completed, assembled, pondered,
built, produced, made, contrived, conceived, actualized, intended, congregated,
designed, pointed at, planned, aimed, calculated, steered, directed, laughed at,
proposed, trained, and engineered by Thomas Schoberl and Laura DeHollander.
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 Hints
1-seq (x, x, 0, 10, 1)
2-binompdf (10, .1, L1)
each P(x =…)
L1
L2 will calculate
3- Turn off any Y= “stuff”
4-[STAT]
L1
1.
2.
3.
4.
5.
6.
L2
.34868
.38742
.19371
.0574
…
…
L3
[PLOT]… histogram
X list = L1
Y list = L2
5- Set window
X[-.5, 10.5]
Y[-.1, .45]