AMS 311
Lecture 14
March 9, 2000
Theorem 4.1. If X is a constant random variable, that is, if P( X  c)  1 for a constant c,
then E ( X )  c.
Law of the unconscious statistician
Theorem 4.2. Let X be a discrete random variable with set of possible values A and
probability function p(x), and let g be a real valued function. Then g(X) is a random
variable with
E[ g( X )]   g( x) p( x).
xA
Corollary Let X be a discrete random variable; g1 , g 2 , , g n be real valued functions, and
let  1 ,  2 , ,  n be real numbers. Then
E[ 1 g1 ( X )   2 g 2 ( X )    n g x ( X )]   1 E[ g1 ( X )]   2 E[ g2 ( X )]   n E[ gn ( X )].
Variances
Definition
Let X be a discrete random variable with a set of possible values A, probability function
p(x), and E ( X )   . Then  X and var( X ) , called the standard deviation and variance of
X, respectively, are defined by
X 
E[( X   ) 2 ] and var( X )  E[( X   ) 2 ].
Example: Find variance of X, the number of red cards identified by the Great Carsoni.
Theorem 4.3.
var( X )  E ( X 2 )  [ E ( X )]2 .
Theorem 4.4.
Let X be a discrete random variable with a set of possible values A, and mean . Then
var( X )  0 if an only if X is a constant with probability 1.
Theorem 4.5.
Let X be a discrete random variable; then for constants a and b we have that
var(aX  b)  a 2 var( X ), and  aX  b  | a| X .
Definition
Let X and Y be two random variables and  be a given point. If for all t>0,
P(|Y   |  t )  P(| X   |  t ), then we say that X is more concentrated about  than is Y.
Theorem 4.6. Suppose that X and Y are two random variables with E ( X )  E (Y )   . If
X is more concentrated about  than is Y, then var( X )  var(Y ).
Case Study
A casino offers the opportunity to play roulette to a gambler. There are 38 equally
probable slots on the wheel. Of these 18 are red, so that the gambler calculates the
18
probability of a red outcome as
=0.4737. The gambler has the following strategy.
38
He will bet $5 on red on the first play. If he wins, he will stop the sequence. He has won
$5. If he loses, he will bet $10 on the second play. If he wins, he will collect $10 (for a
total gain of $5). If he loses on the second play, he will stop. His total gain was -$15.
What are the expected value and variance of his winnings?
Solution:
1. Create the sample space. It has three end points: a. red on the first play; black on
the first play and red on the second play; black on both plays.
2. Now find the probability of each point in the sample space: red on first play
occurs with probability 0.4737; black on first play and read on second play occurs
with probability 0.4737(1-0.4737)=0.2493; black on both trials is
0.52632=0.2770. Check that the probabilities sum to one.
3. Now define the random variable W, the gambler’s total winnings in a sequence for
each sequence in the sample space. The gambler wins $5 total for the first
sequence, $5 total for the second sequence, and -$15 for the third.
4. Calculate E (W )  5(0.4737)  5(0.2493)  (  15)(0.2770)   $0.54.
5. Calculate
var(W )  E (W 2 )  [ E (W )]2  25(0.4737)  25(0.2493)  225(0.2770)  (  0.54) 2
 80.40  0.29  8011
. .
These are the results. The interpretation is more subjective. The expected winnings are
negative; that is, the game is not advantageous to the gambler. The variance is high,
($8.95)2. The gambler has been put on notice that the strategy may be problematic. One
concern is whether there is a small probability of a high loss. Many clients of risk
management specialists have an aversion to such a system. Compare and contrast the
risks and rewards of this game with the strategy of the humble honey bee.
Extra credit problem: generalize the strategy. A roulette table has a maximum amount of
bet that it will accept, say $2500. Calculate the expected winning and variance of winning
for a strategy: bet $5 on red at first play; if loss, bet $10 on red on second play; if two
losses in a row, bet $20 on red on third play; and so on.
Problem: Let X be an indicator variable; that is, it takes the value 1 with probability p and
the value 0 with probability 1-p. What is the expected value and variance of X?
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AMS 311, Two Problem Quiz, March 9, 2000
Name:
Directions: work each problem in the space underneath the question. You may use one
book of notes and your calculator. You may not use any other assistance.
1. Let E, F, and G be three events. Determine which of the following statement are
correct and which are incorrect. Justify your answers:
a. ( E  EF )  F  E  F.
b. F C G  E C G  G( F  E ) C .
c. ( E  F ) C G  E C F C G.
2. In a bridge game, each of the four players gets 13 random cards. What is the
probability that every player has an ace?
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AMS 311, Two Problem Quiz, March 9, 2000
Name:
Directions: work each problem in the space underneath the question. You may use one
book of notes and your calculator. You may not use any other assistance.
1. Let E, F, and G be three events. Determine which of the following statement are
correct and which are incorrect. Justify your answers:
a. F C G  E C G  G( F  E ) C .
b. ( E  F ) C G  E C F C G.
c. EF  EG  FG  E  F  G.
2. In a bridge game, each of the four players gets 13 random cards. What is the
probability that there is a player who has all four aces?
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AMS 311, Two Problem Quiz, March 9, 2000
Name:
Directions: work each problem in the space underneath the question. You may use one
book of notes and your calculator. You may not use any other assistance.
1. Let E, F, and G be three events. Determine which of the following statement are
correct and which are incorrect. Justify your answers:
a. EF  EG  FG  E  F  G.
b. ( E  EF )  F  E  F.
c. F C G  E C G  G( F  E ) C .
2. In a bridge game, each of the four players gets 13 random cards. What is the
probability that at least one player does not have an ace?
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AMS 311, Two Problem Quiz, March 9, 2000
Name:
Directions: work each problem in the space underneath the question. You may use one
book of notes and your calculator. You may not use any other assistance.
1. Let E, F, and G be three events. Determine which of the following statement are
correct and which are incorrect. Justify your answers:
a. EF  EG  FG  E  F  G.
b. ( E  EF )  F  E  F.
c. F C G  E C G  G( F  E ) C .
2. In a bridge game, each of the four players gets 13 random cards. What is the
probability that one of the four players has three aces?