11/29/2010

Probability Generating Functions

• Suppose that the R.V. X can take on values in the set of nonnegative integers {0, 1, … }.  The probability generating function (p.g.f.) for X is g(t)=E[t^X].
• Examples
  • Find the p.g.f. for a R.V. that records the results of the flip of a balanced coin.
  • Use the p.g.f. to find the probability distribution of a R.V. that records the number of heads in four flips of a fair coin.
  • Use the p.g.f. to find the probability distribution for the number of spots on the upward facing sides of 8 dice.

Chebychev's Inequality

• Example - Suppose the length of time for one individual to be served at a cafeteria is a R.V. that follows an exponential distribution with mean 4.
  • What can we assert about the length of time it takes a person to be served at the cafeteria, if we use Chebychev's inequality with k=1.5?
• What is the exact probability for this event?
• What does this tell you about the inequality?

The Central Limit Theorem

Please read Sections 8.4 and 8.5 for class on Wednesday.