Transcript Lecture 5 - Statistics

Today
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Today: Begin Chapter 3
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Reading:
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Covered 2.1-2.5 from Chapter 2
Please read Chapter 3
Suggested Problems: 3.2, 3.9, 3.12, 3.20, 3.23, 3.24, 3R5, 3R9
Example
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Have 50 men, each 26 years old
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Have 50 women, each 28 years old
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What is the average age of the 100 people?
Example
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Have 500 men, each 26 years old
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Have 50 women, each 28 years old
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What is the average age of the 100 people?
Example
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Recall the game show, Let’s Make a Deal
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A contestant had won $9,000 in prizes, and was offered to exchange
the gifts for whatever lay behind one of three doors
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Behind one door was a $20,000 prize and behind the others were
$5,000 and $2,000 prizes, respectively
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Should the contestant make the exchange?
Expectation (The Mean)
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If X is a discrete random variable with probability mass function f(x),
the expected value (or mean value) is
  E ( X )   xi f ( xi )
i
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Provided the sum is absolutely convergent (if there are infinitely
many values x1,x2,… )
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Idea, the mean is the weighted average of the possible values of X
Example
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Recall the game show, Let’s Make a Deal
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A contestant had won $9,000 in prizes, and was offered to exchange
the gifts for whatever lay behind one of three doors
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Behind one door was a $20,000 prize and behind the others were
$5,000 and $2,000 prizes, respectively
•
Should the contestant make the exchange?
Example (True Story)
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When Derek was a graduate student in Vancouver, parking was
$9.00/day
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If you parked illegally, the ticket was $10.00
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Derek discovered that he got a ticket about 50% of the time
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Which is the better strategy:
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Pay $9.00/day
Park illegally
Properties of Expectation
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For random variables X and Y,
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E(c)=c, where c is a constant
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E(cX)=cE(X), where c is a constant
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E(X+Y) = E(X) + E(Y)
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E(aX+bY+c) = aE(X) + bE(Y)+c, where a,b, and c are constants
Example
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Two dice are rolled – one red and one green
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Let X be the outcome of the red die and Y be the outcome of the
green die
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Find E(X+Y)
Conditional Mean
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Conditional distributions also have means
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The mean will be conditional on the value of another random
variable
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The conditional mean of Y give X=x is
Y | x
f ( x, y i )
 E (Y | X  x)   yi f ( yi | X  x)   yi
f X ( x)
i
i
Example
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Three digits are picked at random, without replacement, from 1,2, …, 8
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Let Y denote the largest digit and X denote the smallest
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Find the probability function for Y
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Find E(Y)
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Find probability function for Y given X=3
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Find E(Y|X=3)
Expected Value of a Function
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After observing a random variable, often interested in some function
of the random variable
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The mean value of a function, g, of a random variable X is:
E ( g ( X ))   g ( x) f ( x)
x
Example (3.1)
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Consider a random variable X, with probability function f below:
Find E(X2)
x
0
1
2
3
4
f(x)
.1
.3
.3
.1
.2
Example (3.10)
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Recall the game show, Let’s Make a Deal
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A contestant had won $9,000 in prizes, and was offered to exchange the gifts
for whatever lay behind one of three doors
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Behind one door was a $20,000 prize and behind the others were $5,000 and
$2,000 prizes, respectively
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Suppose that the contestant owes a murderous loan shark $9000, due the
next day
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Can use a utility function to help express thenotion of expectation in this
situation
Expected Values and Joint Distributions
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Some useful relations:
E ( f ( y | X ))  fY ( y )
and
E ( E (Y | X ))  E (Y )
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How do we get these?
Expected Values and Joint Distributions
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When random variables X and Y are independent,
E ( g ( X )h(Y ))  E ( g ( X )) E (h(Y ))
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How did we get this?