Binomial and Geometric Distributions

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Binomial and Geometric Distributions

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It consists of a fixed number of observations called trials. ... Let's look at the case of five customers, a binominal experiment with five trials. ... – PowerPoint PPT presentation

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Title: Binomial and Geometric Distributions


1
Binomial and Geometric Distributions
2
Properties of Binomial Experiments
  • It consists of a fixed number of observations
    called trials.
  • Each trial can result in one of only two mutually
    exclusive outcomes success (S) and failure (F)
  • Outcomes of different trials are independent.
  • The probability that a trial results in S is the
    same for each trial.

3
Binomial Random Variable
  • The binomial random variable x is defined as
  • x number of successes observed when an
    experiment is performed
  • The probability distribution of x is called the
    binomial probability distribution.

4
Example
  • A few days ago we studied the purchasing
    characteristics of customers shopping for a hot
    tub. We considered x number among four
    customers who selected an electric (as opposed to
    gas) hot tub.
  • This is a binomial experiment with
  • number of trials 4 P(success) P(E)0.4

5
New Example
  • Lets look at the case of five customers, a
    binominal experiment with five trials.
  • Possible x values are 0, 1, 2, 3, 4, 5.
  • How many possible outcomes are there? (Remember
    the fundamental counting principle?
  • There are 32 possible outcomes, and five of them
    yield x1.
  • SFFFF FSFFF FFSFF FFFSF FFFFS

6
Example (cont)
  • Lets take a look at the probability for the
    outcomes x1 (SFFFF, FSFFF, FFSFF, )
  • The calculation will be the same for any outcome
    with only one success (x1).

7
Example (cont)
8
Example (cont)
  • What about for x2. How many ways can we select
    two from among the five trials to be the Ss?
  • What is the probability of P(SSFFF)?

9
Example (cont)
  • What is p(2) if we know
  • Remember

10
The Binomial Distribution
  • Ready for this
  • Let
  • n number of independent trials in a binomial
    experiment
  • p constant probability that any particular
    trial results in a success

11
Example (Revisited)
  • Lets go back to our 5 customers and look at p(2)
    using our new formula.
  • We said.
  • Does that fit with our formula?
  • n 5 x2 p.4

12
Next Example
  • 60 of all watches sold by a large discount store
    have a digital display and the rest have analog.
    The type of watch purchased by the next 12
    customers will be noted.
  • x number of watches that have a digital
    display.

13
Next Example
  • What is the probability that 4 watches are
    digital? Go ahead and try it now

14
Using Appendix Table X
  • To find p(x) for any particular value of x,
  • Locate the part of the table corresponding to
    your value of n (5, 10, 15, 20, or 25)
  • Move down to the row labeled with your value of
    x.
  • Go across to the column headed by the specified
    value of p
  • Try it now lets do n20 and p 0.8
  • Find p(15).

15
Homework
  • 7.42
  • 7.43 (cant use table)
  • 7.44 (cant use table)
  • 7.45 (use table)
  • 7.47
  • 7.48
  • 7.49
  • 7.50
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