Probability Models for Distributions of Discrete Variables

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Probability Models for Distributions of Discrete
Variables
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Daily of calls to a fire department x of
calls p(x) relative frequency of x calls
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Multiply each value by its relative frequency /
proportion / probability Sum the products
Mean 1.80
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The Variance is obtained in a similar fashion.
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• The mean and standard deviation (and percentiles
  for continuous data) are not systematically
  dependent on the size of the data set. They just
  depend on the likelihood of the various possible
  values.

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Populations / Samples

• A population is a collection of all units of
  interest.
• Example All students at SUNY Oswego
• A sample is a collection of units drawn from the
  population.
• Example Students in the class
• Example A random sample of 50 students.

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Probability Models and Populations

• The difference between a population and a sample
  is conceptual. For discrete data, the two
  (population and sample) can be summarized the
  same way (for instance, as a table of values and
  accompanying relative frequencies).
• A probability distribution (or model) for a
  discrete variable is a description of values,
  with each value accompanied by a probability.
• Suppose a random selection of a single unit is
  performed over and over (technically forever)
  The probability is the long term relative
  frequency of occurrence of that value.
• The probability associated with a value tells you
  its relative frequency of occurrence over all
  possible ways the phenomena could take place.
• Because a probability describes how often over
  all possible outcomes, a probability is a
  population relative frequency.

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• A probability distribution for a discrete
  variable is tabulated with a set of values, x and
  probabilities, p(x).

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• A probability distribution for a discrete
  variable is tabulated with a set of values, x and
  probabilities, p(x).

Probabilities Must be nonnegative.
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• A probability distribution for a discrete
  variable is tabulated with a set of values, x and
  probabilities, p(x).

Probabilities Must be nonnegative. Must sum to
1. Within rounding error.
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• The mean ? of a probability distribution is the
  mean value observed for all possible outcomes of
  the phenomena.
• Formula
• Example
• ? denotes population mean

SUM symbol
Greek letter myou
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• The standard deviation ? of a probability
  distribution is the standard deviation of the
  values observed for all possible outcomes of the
  phenomena.
• Formula
• Example
• ? denotes population standard deviation

Greek letter sigma
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• If you have discrete data that constitutes a
  random sample of size n, this formula is adjusted
  by a multiplication factor.
• Formula
• There are two SD buttons on your calculator.
  (Neither works for discrete data tabulated by
  relative frequency both require data unit by
  unit.) One computes a population SD and the other
  the sample SD. This does the same thing your
  calculators standard deviation button does.

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Probability Distributions and Populations

• For the most part, the exact size of a population
  doesnt matter Most populations are very large
  and large enough to meet the criteria for methods
  we use. There are exceptions
• Dont use Math 158 methods when the population
  size N is not at least 20 times the sample size
  n.
• N/n should be at least 20.

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• Here is the probability distribution for the
  number of diners seated at a table in a small
  café.

a) Fill in the blank
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
a) Fill in the blank
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Start by computing xp(x)
for each row.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Start by computing xp(x)
for each row.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Start by computing xp(x)
for each row.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Start by computing xp(x)
for each row.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Start by computing xp(x)
for each row.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Sum these.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the mean ? Sum these. ? 3.00
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Start by
computing ( x ? ) 2 p(x) for each row.
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Start by
computing ( x ? )2 p(x) for each row. ? 3
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Start by
computing ( x ? ) 2 p(x) for each row. ? 3
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Start by
computing ( x ? ) 2 p(x) for each row. ? 3
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Start by
computing ( x ? ) 2 p(x) for each row. ? 3
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Start by
computing (x ? ) 2 p(x) for each row. ? 3
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Sum these
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Here is the probability distribution for the
number of diners seated at a table in a small
café.
b) Determine the standard deviation ? Sum
these Variance 1.00 SD ? 1.00
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Given the probability distribution for the number
of diners seated at a table in a small café.
c) Explain how one could obtain the mean and
standard deviation inputting data and using the
stats buttons or functions on a calculator or
computer. ANS Sit outside the café and watch
customers. Every time a table is occupied, enter
the of people into the computer. After a large
of repeats, compute mean SD.
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Given the probability distribution for the number
of diners seated at a table in a small café.
In fact, after a large number of observations of
units, the relative frequencies you observe will
match the probabilities if not, then perhaps
this is not the proper probability distribution?
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Optional Application

• This framework makes it possible to obtain fairly
  good approximations to means and standard
  deviations from a histogram of continuous data.

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Example

• Here are waiting times between student arrivals
  in a class. There are 21 students (20 waits).

Approximate the mean and median. How do they
compare?
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Example Mean

• For each class, determine its frequency and
  corresponding midpoint.

Frequency 10 Midpoint 5
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Example Mean

• Tabulate frequencies and midpoints.

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Example Mean

• Tabulate frequencies and midpoints.

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Example Mean

• Obtain relative frequencies.

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Example Mean

• Obtain relative frequencies.

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Example Mean

• Proceed with the formula

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Example Mean

• Proceed as a discrete population distribution.

Mean
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Example Mean

• Proceed as a discrete population distribution.

Mean ? 14.00
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Example Median

• Find the value with 50 below and 50 above.

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Example Median

• Obtain relative frequencies.

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Example Median

• Find the value with 50 below and 50 above.

10 of 20 50 below 10 Median ? 10.00 Mean ?
14.00 Range ? 44 S.D. ? 11
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Example Data / Exact Values

• 1.3 1.9 1.9 2.5 2.6 3.0 3.6
  3.7 5.9 9.7 10.4 10.6 11.2 13.5
  15.9 21.4 27.5 29.8 33.6 43.5
• Approximations Actual Values
• Median ? 10.0.05 Median
• Mean ? 14.0 Mean
• Range ? 44 Range
• SD ? 11 SD

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Example Data / Exact Values

• 1.3 1.9 1.9 2.5 2.6 3.0 3.6
  3.7 5.9 9.7 10.4 10.6 11.2 13.5
  15.9 21.4 27.5 29.8 33.6 43.5
• Approximations Actual Values
• Median ? 10.0.05 Median 10.05
• Mean ? 14.0 Mean 12.68
• Range ? 44 Range 42.2
• SD ? 11 SD 12.31

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• x children in randomly selected college
  students family.

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• x children in randomly selected college
  students family.
• 0.2194 21.94 of all college students come from
  a 1 child family.

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• To determine the mean, multiply values by
  probabilities,
• x?p(x)
• and sum these.
• 55/10 5.50 is not the mean
• 1.000/10 0.10 is not the mean

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• To determine the variance, multiply squared
  deviations from the mean by probabilities,
• (x ?)2?p(x)
• and sum these.

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• The standard deviation is the square root of the
  variance.
• Examining the data set consisting of of
  children for all students The mean is 2.743 the
  standard deviation is 1.468.

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5)

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5)

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5)

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5)

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5)

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5) 0.0317
• 0.0124
• 0.0043
• 0.0005
• 0.0003

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5) 0.0317
• 0.0124
• 0.0043
• 0.0005
• 0.0003
• 0.0492

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• Determine the probability a student is from a
  family with more than 5 siblings.
• P(x gt 5) 0.0492
• 4.92 of all college students come from families
  with more than 5 children (they have 4 or more
  brothers and sisters).

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• Determine the probability a student is from a
  family with at most 3 siblings.
• P(x ? 3) 0.2194
• 0.2806
• 0.2329
• 0.7329

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• Determine the probability a student is from a
  family with at least 7 siblings.
• P(x ? 7) 0.0124
• 0.0043
• 0.0005
• 0.0003
• 0.0175
• Good idea Take the reciprocal of a small
  probability
• 1/.0175 57.1 ? 1 in 57 students

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• Determine the probability a student is from a
  family with fewer than 5 siblings.
• P(x lt 5) 0.2194
• 0.2806
• 0.2329
• 0.1442
• 0.8771

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• at most 3 at least 7
• ?? ??
• less than or equal to 3 greater than or equal to
  7
• ?? ??
• no more than 3 no less than 7
• ?? ??
• x ? 3 x ? 7