The Normal Distribution

1
The Normal Distribution In Reverse

• Example
• Given a normal distribution with µ 40 and s
  6, find the value of X for which 45 of the area
  under the normal curve is to the left of X.
• If P(Z lt k) 0.45,
• k ___________
• Z _______
• X _________

2
Normal Approximation to the Binomial

• If n is large and p is not close to 0 or 1,
• or
• If n is smaller but p is close to 0.5, then the
  binomial distribution can be approximated by the
  normal distribution using the transformation
• Look at example 6.15, pg. 162

3
Continuous Probability Distributions

• Many continuous probability distributions,
  including
• Uniform
• Normal
• Gamma
• Exponential
• Chi-Squared
• Lognormal
• Weibull

4
Gamma Exponential Distributions

• Recall the Poisson Process
• Number of occurrences in a given interval or
  region
• Memoryless process
• Sometimes were interested in the time or area
  until a certain number of events occur.
• For example
• An average of 2.7 service calls per minute are
  received at a particular maintenance center. The
  calls correspond to a Poisson process.
• What is the probability that up to a minute will
  elapse before 2 calls arrive?
• How long before the next call?

5
Gamma Distribution

• The density function of the random variable X
  with gamma distribution having parameters a
  (number of occurrences) and ß (time or region).
• x gt 0.
• µ aß
• s aß2

6
Exponential Distribution

• Special case of the gamma distribution with a
  1.
• x gt 0.
• Describes the time until or time between Poisson
  events.
• µ ß
• s ß2

7
Example

• An average of 2.7 service calls per minute are
  received at a particular maintenance center. The
  calls correspond to a Poisson process.
• What is the probability that up to a minute will
  elapse before 2 calls arrive?
• ß ________ a ________
• P(X 1) _________________________________
• How long before the next call?

8
Your turn

• Look at problem 2, page 174.

9
Updated homework

• Todays assignment (due Wednesday)
• 12, 13, 15, pp. 157-158
• Add problems 3 8, pg. 164
• Wednesdays assignment (due Friday)
• Add problem 8, pg. 174 to the assignment