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Chapter 5: Probability Distributions (Discrete Variables)

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Title: Chapter 5: Probability Distributions (Discrete Variables)


1
Chapter 5 Probability Distributions(Discrete
Variables)
2
Chapter Goals
  • Combine the ideas of frequency distributions and
    probability to form probability distributions.
  • Investigate discrete probability distributions
    and study measures of central tendency and
    dispersion.
  • Study the binomial random variable.

3
5.1 Random Variables
  • Bridge between experimental outcomes and
    statistical analysis.
  • Each outcome in an experiment is assigned to a
    number.
  • This suggests the idea of a function.

4
  • Random Variable A variable that assumes a unique
    numerical value for each of the outcomes in the
    sample space of a probability experiment.
  • Note
  • 1. Used to denote the outcomes of a probability
    experiment.
  • 2. Each outcome in a probability experiment is
    assigned to a unique value.
  • 3. Illustration

5
  • Examples of random variables
  • 1. Let the number of computers sold per day by a
    local merchant be a random variable. Integer
    values ranging from zero to about 50 are possible
    values.
  • 2. Let the number of pages in a mystery novel at
    a bookstore be a random variable. The smallest
    number of pages is 125 while the largest number
    of pages is 547.
  • 3. Let the time it takes an employee to get to
    work be a random variable. Possible values are
    15 minutes to over 2 hours.
  • 4. Let the volume of water used by a household
    during a month be a random variable. Amounts
    range up to several thousand gallons.
  • 5. Let the number of defective components in a
    shipment of 1000 be a random variable. Values
    range from 0 to 1000.

6
  • Discrete Random Variable A quantitative random
    variable that can assume a countable number of
    values.
  • Intuitively, a discrete random variable can
    assume values corresponding to isolated points
    along a line interval. That is, there is a gap
    between any two values.
  • Note Usually associated with counting.
  • Continuous Random Variable A quantitative random
    variable that can assume an uncountable number of
    values.
  • Intuitively, a continuous random variable can
    assume any value along a line interval, including
    every possible value between any two values.
  • Note Usually associated with a measurement.

7
  • Example Determine whether the following random
    variables are discrete or continuous.
  • 1. The barometric pressure at 1200 PM.
  • 2. The length of time it takes to complete a
    statistics exam.
  • 3. The number of items in the shopping cart of
    the person in front of you at the checkout line.
  • 4. The weight of a home grown zucchini.
  • 5. The number of tickets issued by the PA State
    Police during a 24 hour period.
  • 6. The number of cans of soda pop dispensed by a
    machine placed in the Mathematics building on
    campus.
  • 7. The number of cavities the dentist discovers
    during your next visit.

8
5.2 Probability Distributions of a Discrete
Random Variable
  • Need a complete description of a discrete random
    variable.
  • This includes all the values the random variable
    may assume and all of the associated
    probabilities.
  • This information may be presented in a variety of
    ways.

9
  • Probability Distribution A distribution of the
    probabilities associated with each of the values
    of a random variable. The probability
    distribution is a theoretical distribution it is
    used to represent populations.
  • Note
  • 1. The probability distribution tells you
    everything you need to know about the random
    variable.
  • 2. The probability distribution may be presented
    in the form of a table, chart, function, etc.
  • Probability Function A rule that assigns
    probabilities to the values of the random
    variable.

10
  • Example The number of people staying in a
    randomly selected room at a local hotel is a
    random variable ranging in value from 0 to 4.
    The probability distribution is known and is
    given in various forms below.
  • Note
  • 1. This chart implies the only values x takes on
    are 0, 1, 2, 3, and 4.
  • 2.

11
  • A line representation of the Hotel Room
    probability distribution

12
  • A histogram may be used to present a probability
    distribution.
  • A histogram for the Hotel Room probability
    distribution

13
  • Note
  • 1. The histogram of a probability distribution
    uses the physical area of each bar to represent
    its assigned probability.
  • 2. In the Hotel Room probability distribution
    the width of each bar is 1, so the height of each
    bar is equal to the assigned probability, which
    is the area of each bar.
  • 3. The idea of area representing probability is
    important in the study of continuous random
    variables.

14
  • Reminder Every probability function must satisfy
    the two basic properties of probability.
  • 1. The probability assigned to each value of the
    random
  • variable must be between 0 and 1, inclusive
  • 2. The sum of the probabilities assigned to all
    the values of
  • the random variable must equal 1

15
5.3 Mean and Variance of a Discrete Probability
Distribution
  • Describe the center and spread of a population.
  • m, s, s2 population parameters.
  • Population parameters are usually unknown values
    (we would like to estimate).

16
  • Note
  • 1. is the mean of the sample.
  • 2. s2 and s are the variance and standard
    deviation of the sample.
  • 3. , s2, and s are called sample statistics.
  • 4. m (lowercase Greek letter mu) is the mean of
    the population.
  • 5. s2 (sigma squared) is the variance of the
    population.
  • 6. s (lowercase Greek letter sigma) is the
    standard deviation of the population.
  • 7. m, s2, and s are called population parameters.
    (A parameter is a constant. m, s2, and s are
    typically unknown values.)

17
  • Mean of a Discrete Random Variable
  • The mean, m, of a discrete random variable x is
    found by multiplying each possible value of x by
    its own probability and then adding all the
    products together.
  • Note
  • 1. The mean is the average value of the random
    variable, what happens on average.
  • 2. The mean is not necessarily a value of the
    random variable.

18
  • Variance of a Discrete Random Variable
  • Variance, s2, of a discrete random variable x is
    found by multiplying each possible value of the
    squared deviation from the mean, (x - m)2, by its
    own probability and then adding all the products
    together.
  • Standard Deviation of a Discrete Random Variable
  • The positive square root of the variance.

19
  • Example The number of standby passengers who get
    seats on a daily commuter flight from Boston to
    New York is a random variable, x, with
    probability distribution given below (in an
    extensions table). Find the mean, variance, and
    standard deviation.

20
  • Solution
  • Using the formulas for mean, variance, and
    standard deviation
  • Note 1.55 is not a value of the random variable
    (in this case). It is only what happens on
    average.

21
  • Example The probability distribution for a
    random variable x is given by the probability
    function
  • Find the mean, variance, and standard deviation.
  • Solution
  • Find the probability associated with each value
    by using the probability function.

22
  • Use an extensions table to find the population
    parameters.

23
5.4 The Binomial Probability Distribution
  • One of the most important discrete distributions.
  • Based on a series of repeated trials whose
    outcomes can be classified in one of two
    categories success or failure.
  • Distribution based on a binomial probability
    experiment.

24
  • Binomial Probability Experiment
  • An experiment that is made up of repeated trials
    that possess the following properties
  • 1. There are n repeated independent trials.
  • 2. Each trial has two possible outcomes (success,
    failure).
  • 3. P(success) p, P(failure) q, and p q
    1
  • 4. The binomial random variable x is the count of
    the number of successful trials that occur x
    may take on any integer value from zero to n.

25
  • Note
  • 1. Properties 1 and 2 are the two basic
    properties of any binomial experiment.
  • 2. Property 3 concerns the algebraic notation for
    each trial.
  • 3. Property 4 concerns the algebraic notation for
    the complete experiment.
  • 4. Both x and p must be associated with
    success.
  • 5. Independent trials mean that the result of one
    trial does not affect the probability of success
    of any other trial in the experiment. The
    probability of success remains constant
    throughout the entire experiment.

26
  • Example It is known that 40 of all graduating
    seniors on campus have taken a statistics class.
    Five seniors are selected at random and asked if
    they have taken a statistics class.
  • 1. A trial is asking one student, repeated 5
    times. The trials are independent since the
    probability of taking a statistics class for any
    one student is not affected by the results from
    any other student.
  • 2. Two outcomes on each trial
  • taken a statistics class (success),
  • not taken a statistics class (failure)
  • 3. p P(taken a statistics class) .40
  • q P(not taken a statistics class) .60
  • 4. x number of students who have taken a
    statistics class

27
  • Binomial Probability Function
  • For a binomial experiment, let p represent the
    probability of a success and q represent the
    probability of a failure on a single trial
    then P(x), the probability that there will be
    exactly x successes on n trials is
  • Note
  • 1. The number of ways that exactly x successes
    can occur in n trials
  • 2. The probability of exactly x successes px
  • 3. The probability that failure will occur on the
    remaining (n - x) trials qn - x

28
  • Note
  • The number of ways that exactly x successes can
    occur in a set of n trials is represented by the
    symbol
  • 1. Must always be a positive integer.
  • 2. Called the binomial coefficient.
  • 3. Found by using the formula
  • n! is an abbreviation for n factorial.

29
  • Example According to a recent study, 65 of all
    homes in a certain county have high levels of
    radon gas leaking into their basements. Four
    homes are selected at random and tested for
    radon. The random variable x is the number of
    homes with high levels of radon (out of the
    four).
  • Properties
  • 1. There are 4 repeated trials n 4. The
    trials are independent.
  • 2. Each test for radon is a trial, and each test
    has two outcomes radon or no radon.
  • 3. p P(radon) .65, q P(no radon) .35
  • p q 1
  • 4. x is the number of homes with high levels of
    radon,
  • possible values 0, 1, 2, 3, 4

30
  • The binomial probability function

31
  • Example In a certain automobile dealership, 70
    of all customers purchase an extended warranty
    with their new car. For 15 customers selected at
    random
  • 1. Find the probability that exactly 12 will
    purchase an extended warranty.
  • 2. Find the probability at most 13 will purchase
    an extended warranty.
  • Solution
  • Let x be the number of customers who purchase an
    extended warranty. x is a binomial random
    variable.
  • The probability function associated with this
    experiment

32
  • Probability exactly 12 purchase an extended
    warranty
  • Probability at most 13 purchase an extended
    warranty

33
  • Note
  • 1. The value of many binomial probabilities (n lt
    15 and common values of p) are found in Table 2,
    Appendix B.
  • 2. Minitab has special commands for computing
    binomial probabilities or cumulative
    probabilities.
  • PDF 12
  • Binomial 15 .7.
  • Probability Density Function
  • Binomial with n 15 and p 0.700000
  • x P( X x)
  • 12.00 0.1700
  • 3. Many graphing calculators also have built-in
    functions for computing binomial probabilities
    and cumulative probabilities.

34
  • Notation
  • If x is a binomial random variable with n trials
    and probability of a success p, this is often
    denoted x B(n, p).
  • Example Suppose x is a binomial random variable
    with n 18 and p .35. A convenient notation
    to identify this random variable is x B(18,
    .35).

35
5.5 Mean and Standard Deviation of the Binomial
Distribution
  • Population parameters of the binomial
    distribution help to describe the distribution.
  • Mean and standard deviation indicate where the
    distribution is centered and the spread of the
    distribution.

36
  • The mean and standard deviation of a theoretical
    binomial distribution can be found by using the
    following two formulas
  • Note
  • 1. Mean is intuitive number of trials multiplied
    by the probability of a success.
  • 2. The variance of a binomial probability
    distribution is

37
  • Example Find the mean and standard deviation of
    the binomial distribution when n 18 and p
    .75.
  • Solution
  • n 18, p .75, q 1 - .75 .25
  • The probability function is

38
  • Table of values and probabilities
  • x P( X x)
  • 0.00 0.0000
  • 1.00 0.0000
  • 2.00 0.0000
  • 3.00 0.0000
  • 4.00 0.0000
  • 5.00 0.0000
  • 6.00 0.0002
  • 7.00 0.0010
  • 8.00 0.0042
  • 9.00 0.0139
  • 10.00 0.0376
  • 11.00 0.0820
  • 12.00 0.1436
  • 13.00 0.1988
  • 14.00 0.2130
  • 15.00 0.1704
  • 16.00 0.0958

39
  • Histogram
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