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BCOR 1020 Business Statistics Lecture 10 – February 19, 2008 Overview Chapter 6 – Discrete Distributions Bernoulli Distribution Binomial Distribution Chapter 6 – Bernoulli Distribution Bernoulli Experiments:

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bcor 1020 business statistics

BCOR 1020Business Statistics

Lecture 10 – February 19, 2008

overview
Overview
  • Chapter 6 – Discrete Distributions
    • Bernoulli Distribution
    • Binomial Distribution
chapter 6 bernoulli distribution
Chapter 6 – Bernoulli Distribution

Bernoulli Experiments:

  • A Bernoulli Experiment is a random experiment, the outcome of which can be classified in one of two mutually exclusive and exhaustive ways:
    • One outcome is arbitrarily labeled a “success” (denoted X = 1) and the other a “failure” (denoted X = 0).
  • P(X = 1) = P(success) = p
  • P(X = 0) = P(failure) = 1 – p
  • Note that P(0) + P(1) = (1 – p) + p = 1 and 0 <p< 1.
  • “Success” is usually defined as the less likely outcome so that p < .5 for convenience.
chapter 6 bernoulli distribution4
Chapter 6 – Bernoulli Distribution

Bernoulli Experiments:

Some examples of Bernoulli experiments:

chapter 6 bernoulli distribution5
Chapter 6 – Bernoulli Distribution

The PDF of the Bernoulli Experiment:

The Mean and Variance of the Bernoulli Experiment:

chapter 6 binomial distribution
Chapter 6 – Binomial Distribution

Bernoulli Trials:

  • If we repeat a Bernoulli Experiment several (n) times independently, we have n Bernoulli trials.
    • n Independent Bernoulli experiments
    • P(“success”) = p on each of the n trials
  • Example: Suppose we roll a single die 4 times and observe whether a “1” is rolled each time.
  • This would constitute a set of n = 4 Bernoulli trials with …
  • p = P(“success”) = P(“1” is rolled)
chapter 6 binomial distribution7
Chapter 6 – Binomial Distribution

The Binomial Distribution:

  • If X denotes the number of “successes” observed in n Bernoulli trials, then we say that X has the Binomial distribution with parameters n and p.
  • This is often denoted X~b(n,p)
  • We can determine the following about X based on the values of n and p :
  • The PDF of X, f(x) = P(X = x)
  • The mean of X, m = E(X)
  • The variance of X, s 2 = V(X)
  • The std. deviation of X, s
chapter 6 binomial distribution8
Chapter 6 – Binomial Distribution

The PDF of the Binomial Distribution:

  • To find the PMF of the Binomial distribution, we observe that it consists of n independent Bernoulli distributions, each with parameter p.
  • The probability of observing x “successes” in n Bernoulli trials will be given by the product of …
  • P(x “successes”) and P((n – x) “failures”)
  • And the number of ways in which this combination of successes and failures can occur
chapter 6 binomial distribution9
Chapter 6 – Binomial Distribution

The PDF of the Binomial Distribution:

The mean of the Binomial Distribution:

The variance and standard deviation of the

Binomial Distribution:

chapter 6 binomial distribution11
Chapter 6 – Binomial Distribution

Example: Quick Oil Change Shop

  • It is important to quick oil change shops to ensure that a car’s service time is not considered “late” by the customer.
  • Service times are defined as either late or not late.
  • X is the number of cars that are late out of the total number of cars serviced.
  • Assumptions: - cars are independent of each other - probability of a late car is consistent
chapter 6 binomial distribution12
Chapter 6 – Binomial Distribution

Example: Quick Oil Change Shop

  • Assuming that the probability that a car is late is P(late) = p = 0.10,…
  • For the next n = 10 cars serviced, what are the mean and standard deviation for the number of late cars?
  • What is the probability that exactly 2 of the next n = 10 cars serviced are late (i.e. P(X = 2))?
slide13

Clickers

A real estate agent estimates that her probability of

selling a house is 10%. If she shows houses to 5

customers today, what is the expected number of

houses she will sell?

A = 0.0

B = 0.2

C = 0.5

D = 1.0

slide14

Clickers

A real estate agent estimates that her probability of

selling a house is 10%. If she shows houses to 5

customers today, what is the standard deviation for

the number of houses she will sell?

A = 0.45

B = 0.67

C = 0.90

D = 0.95

slide15

Clickers

A real estate agent estimates that her probability of

selling a house is 10%. If she shows houses to 5

customers today, what is the probability that she

will sell one house?

A = 0.4095

B = 0.3281

C = 0.6719

D = 0.5905

chapter 6 binomial distribution16
Chapter 6 – Binomial Distribution

Binomial Shape:

  • A binomial distribution is
    • skewed right if p < .50,
    • skewed left if p > .50,
    • and symmetric if p = .50
  • Skewness decreases as n increases, regardless of the value of p.
    • For large n, the binomial distribution is symmetric!
  • To illustrate, consider the following graphs:
chapter 6 binomial distribution18

P(X 2) = P(2) + P(3) + P(4)

P(X 2) = 1 – P(X < 2) = 1 – P(0) – P(1)

Chapter 6 – Binomial Distribution

Compound Events:

  • Individual probabilities can be added to obtain any desired event probability.
  • For example, the probability that a sample of 4 patients will contain at least 2 uninsured patients is
  • HINT: What inequality means “at least?”

In this example we can also use the complimentary probability, P(A’) = 1 – P(A):

chapter 6 binomial distribution19
Chapter 6 – Binomial Distribution

Compound Events:

  • We must determine which inequality corresponds to our problem:
  • “fewer than” = “less than” =
  • “at most” = “no more than” =
  • “more than” = “greater than” =
  • “at least” = “no fewer than” =
slide20

Clickers

A real estate agent estimates that her probability of

selling a house is 10%. If she shows houses to 5

customers today, what is the probability that she

will sell at least one house?

A = 0.4095

B = 0.3281

C = 0.6719

D = 0.5905

chapter 6 binomial distribution21
Chapter 6 – Binomial Distribution

Recognizing Binomial Applications:

  • Look for n independent Bernoulli trials with constant probability of success.
    • Recall: A Bernoulli Experiment is one in which there are two mutually exclusive and exhaustive outcomes (“success” or “failure”).

P(“success”) = p

    • If we repeat a Bernoulli Experiment n times independently then we have n Bernoulli trials with

P(“success”) = p.

  • If we are counting the number of “successes” observed in n Bernoulli trials, then we have a Binomial Distribution, X~b(n,p).
slide22

Clickers

In a manufacturing process involving 5 mm bolts,

our supplier guarantees that 95% of the bolts they

sell to us meet specifications. If we select 50 of

these bolts at random and observe the number of

bolts that don’t meet specification, what type of

probability distribution best describes our experiment?

A = Bernoulli Distribution

B = Binomial with n = 50, p = 0.95

C = Binomial with n = 50, p = 0.05

D = None of the above