Today

1
Today

• Today Finish Chapter 3, begin Chapter 4
• Reading
• Have done 3.1-3.5
• Please start reading Chapter 4
• Suggested problems 3.24, 4.2, 4.8, 4.10, 4.33,
  4.42, 4R3, 4R5

2
Some Useful Properties of Mean and Variances

• Expectation of a Sum of Random Variables
• Variance of a Sum of Random Variables

3
Example (3.23)

• X and Y are random variables with the following
  joint distribution
• Find cov(X,Y)
• Find Var(XY)
• Find Var(YX0)

4
Chapter 4 (Some Discrete Random Variables)

• Sometimes we are interested in the probability
  that an event occurs (or does not occur)
• In this case, we are interested in the
  probability of a success (or failure)
• In this case, the the random variable of
  interest, X, takes on one of two possible
  outcomes (success and failure) coded 1 and 0
  respectively
• The probability of a success is P(X1)p
• The probability of a failure is P(X0)1-pq

5
Bernouilli Distribution

• The probability function for a Bernoulli random
  variable is
• Can be written as

x f(x)
1 p
0 q1-p
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Bernouilli Distribution

• Mean
• Variance

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Characteristics of a Bernoulli Process

• Trials are independent
• The random experiment takes on only 2 values
  (X1 or 0)
• The probability of success remains constant

8
Example

• A fair coin
• An unfair coin

9
Binomial Distribution

• More often concerned with number of successes in
  a specified number of trials

10
Example

• A gambler plays 10 games of roulette. What is
  the probability that they break even?

11
Binomial Distribution

• Probability Function

12
Binomial Distribution

• Mean
• Variance

13
Example

• To test a new golf ball, 20 golfers are paired
  together by ability (I.e., there are 10 pairs)
• One golfer in each pair plays with the new golf
  ball and the other with an older variety
• Each pair plays a round of golf together
• Let X be the number of pairs in which the player
  with new ball wins the match

14
Example

• If the new ball performs as well as the old ball,
• What is the distribution of X?
• Find P(Xlt2)
• Find the mean and variance of X

15
Example

• ESP researchers often use Zener cards to test ESP
  ability
• A deck of cards consists of equal numbers of five
  types of cards showing very different shapes
• Some people believe that hypnosis helps ESP
  ability
• A card is randomly sampled from the deck
• A hypnotized person concentrates on the card and
  guesses the shape

16
Example

• 10 students performed this test
• Each student had three guesses
• In total there were 10 correct guesses. Is this
  evidence in favor of ESP ability?

17
Independent Binomials

• If X and Y are independent binomial random
  variables with distributions Bin(n1,p) and
  Bin(n2,p) then XY has a Bin(n1 n2,p)

18
Discrete Uniform Distribution

• Consider a discrete distribution, with a finite
  number of outcomes, where each outcome has the
  same chance of occurring
• Suppose the possible outcomes of a random
  variable, X, are 1,2,n, with equal probability.
  What is the probability function for X

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Discrete Uniform Distribution

• Mean
• Variance

20
Example

• Fair die