Revised Lecture Schedule. Note That the Two Problem Quiz Scheduled for March 15 Has Been

AMS 311, Lecture 12

March 13, 2001

Revised Lecture Schedule. Note that the two problem quiz scheduled for March 15 has been put on March 29.

Mar 13: Finish Chapter 5.

Mar 15:Sections 6.1 to 6.3. Homework Chapter Five due.

Mar 20:Spring Recess

Mar 22: Spring Recess

Mar 27: Sections 7.1 to 7.2. Homework Chapter Six due.

Mar 29:Sections 7.3 to 7.4. Two problem quiz: Transformation of a univariate random variable, moments of a univariate random variable.

Apr 3:Section 7.5, Section 8.1.

Apr 5: Sections 8.2 to 8.3. Homework Chapter Seven due.

Apr 10:Sections 8.4 to 8.7.

Apr 12: Sections 9.1 to 9.2. Two problem quiz: transformation of bivariate random variables, finding a conditional pdf. Homework Chapter Eight due.

Apr 17:Review for Examination 2.

Apr 19:Examination 2

Apr 24:Examination 2 returned, sections 9.3 to 9.5

Apr 26:Sections 10.1 to 10.3. Homework Chapter Nine due.

May 1:Section 10.4 and 10.5.

May 3:Chapter 11. Homework Chapter Ten due.

May 8:Review for final examination

May 10: Final examination, 11:00 am to 1:30 pm.

Probabilistic Modeling

• Define the random variable: discrete or continuous?

Modeling Discrete Random Variables:

• Is there an event that is counted?

If there is a maximum number of events that can occur, consider the Bernoulli random variable. Pmf , expected value and variance.

Compare E(X) to var (X).

If var(X)<E(X), can you use the Binomial distribution to model X. Pmf, expected value, and variance.

Binomial Distribution B(n,p)

Counts number of successes in n total trials. Trials independent and homogeneous.

Let X be a binomially distributed random variable for n trials and probability of success p.

Probability function (Theorem 5.1):

• Is the counted event “rare”?

Probability function:

Mean:

Variance:

Example problem: On average, there are three misprints in every 10 pages of a particular book. If a chapter of the book contains 35 pages, what is the probability that that it has one or more misprints.

Poisson Process: It models the number of events that occur in the time interval between 0 and t. Assumptions are: stationarity, independent increments, orderliness (two events cannot happen simultaneously). Let  be the expected number of events that occur in a unit time interval. Then the number of events that occur in the time interval (0, t] is a Poisson random variable with expectation t.

Example: Suppose that earthquakes of a specified magnitude or greater occur in a country in accordance with a Poisson process at a rate of three per year. What is the probability of no earthquakes in a specified two-year period?

Other Discrete Distributions

Geometric: Time to first failure

Probability function:

Mean Variance

Example Problem: The probability that a Bernoulli trial is a success is 0.05. Describe the distribution of the number of events observed until the first success.

Hypergeometric: remember the Great Carsoni.