Download Lecture 7 - Statistics

Today
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Today: Finish Chapter 3, begin Chapter 4
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Reading:
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–
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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
Some Useful Properties of Mean and Variances
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Expectation of a Sum of Random Variables:
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Variance of a Sum of Random Variables:
Example (3.23)
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X and Y are random variables with the following joint distribution
X
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Find cov(X,Y)
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Find Var(X+Y)
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Find Var(Y|X=0)
1
2
0
.2
0
Y
1
.1
.2
2
.3
.2
Chapter 4 (Some Discrete Random Variables)
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Sometimes we are interested in the probability that an event occurs (or does
not occur)
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In this case, we are interested in the probability of a success (or failure)
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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
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The probability of a success is P(X=1)=p
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The probability of a failure is P(X=0)=1-p=q
Bernouilli Distribution
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The probability function for a Bernoulli random variable is:
x f(x)
1 p
0 q=1-p
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Can be written as:
Bernouilli Distribution
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Mean:
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Variance:
Characteristics of a Bernoulli Process
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•
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Trials are independent
The random experiment takes on only 2 values (X=1 or 0)
The probability of success remains constant
Example
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A fair coin:
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An unfair coin:
Binomial Distribution
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More often concerned with number of successes in a specified
number of trials
Example
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A gambler plays 10 games of roulette. What is the probability that
they break even?
Binomial Distribution
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Probability Function:
Binomial Distribution
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Mean:
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Variance
Example
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To test a new golf ball, 20 golfers are paired together by ability (I.e.,
there are 10 pairs)
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One golfer in each pair plays with the new golf ball and the other
with an older variety
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Each pair plays a round of golf together
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Let X be the number of pairs in which the player with new ball wins
the match
Example
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If the new ball performs as well as the old ball,
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What is the distribution of X?
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Find P(X<=2)
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Find the mean and variance of X
Example
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ESP researchers often use Zener cards to test ESP ability
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A deck of cards consists of equal numbers of five types of cards
showing very different shapes
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Some people believe that hypnosis helps ESP ability
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A card is randomly sampled from the deck
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A hypnotized person concentrates on the card and guesses the shape
Example
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10 students performed this test
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Each student had three guesses
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In total there were 10 correct guesses. Is this evidence in favor of
ESP ability?
Independent Binomials
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If X and Y are independent binomial random variables with
distributions Bin(n1,p) and Bin(n2,p) then X+Y has a Bin(n1+ n2,p)
Discrete Uniform Distribution
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Consider a discrete distribution, with a finite number of outcomes,
where each outcome has the same chance of occurring
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Suppose the possible outcomes of a random variable, X, are 1,2,…n,
with equal probability. What is the probability function for X
Discrete Uniform Distribution
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Mean:
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Variance
Example
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Fair die: