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Math 160
Intro to Statistics
OCC
Section 5.2 – The Mean and Standard Deviation of a Discrete Random Variable
Mean of a Discrete Random Variable
The mean of a discrete random variable X is denoted by X or, when no confusion will arise, simply . It is
defined by
   xP ( X  x) or    xP( x)
The terms expected value and expectation are commonly used in place of the term mean.
Interpretation of the Mean of a Random Variable
In a large number of independent observations of a random variable X, the average value of those observations
will approximate equal the mean, , of X; The larger the number of observations, the closer the average tends to
be .
Standard Deviation of a Discrete Random Variable
The standard deviation of a discrete random variable X is denoted by X or, when no confusion will arise, simply
. It is defined as
  ( x   ) 2 P( X  x)
or
  ( x   )2 P( x)
The standard deviation of a discrete random variable X can also be obtained from the computing formula
   x 2 P( X  x)   2
or
   x 2 P( x)   2
 Exercises:
1)
What is the mean number of siblings for Exercise #1 from the handout in Section 5.1?
x
P(X = x)
x·P(X = x)
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Math 160
Intro to Statistics
OCC
Section 5.2 – The Mean and Standard Deviation of a Discrete Random Variable
 Exercises:
2)
What is the standard deviation for the number of siblings for Exercise #1 from the handout in
Section 5.1?
x2
x
3)
P(X = x)
x2·P(X = x)
Roulette is a common game of gambling found in Casinos. An American roulette wheel contains 38
numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally
likely to land on any of the 38 numbers. Suppose that you bet $1 on red. If the ball lands on a red
number, you win $1; otherwise you lose your $1. Let X be the amount you win on your $1 bet. Then X
is a random variable whose probability distribution is as follows:
x
1
–1
P(X = x)
0.474
0.526
a)
Verify that the probability distribution is correct.
b)
Find the expected value of the random variable X.
c)
On average, how much will you lose per play?
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Math 160
Intro to Statistics
OCC
Section 5.2 – The Mean and Standard Deviation of a Discrete Random Variable
 Exercises:
4)
Hosted by Howie Mandel, "Deal or No Deal" was a popular game show that debuted in 2000. The game
revolves around the opening of a set of numbered briefcases, each of which contains a different prize (cash or
otherwise). The contents (i.e., the values) of all of the cases are known at the start of the game, but the specific
location of any prize is unknown. The contestant
claims (or is assigned) a case to begin the game.
The case's value is not revealed until the
conclusion of the game. The following graphic to
the right shows the 26 different
denominations.The contestant then begins
choosing cases that are to be removed from play.
The amount inside each chosen case is
immediately revealed; by process of elimination,
the amount revealed cannot be inside the case the
contestant initially claimed (or was assigned).
Throughout the game, after a predetermined
number of cases have been opened, the banker
offers the contestant an amount of money and/or
prizes to quit the game, the offer based roughly
on the amounts remaining in play and the
contestant's demeanor, the bank tries to 'buy' the
contestant's case for a lower price than what's
inside the case. The player then answers the
titular question, choosing:
"Deal", accepting the offer presented and ending the game, or
"No Deal", rejecting the offer and continuing the game.
This process of removing cases and receiving offers continues, until either the player accepts an offer to 'deal',
or all offers have been rejected and the values of all unselected cases are revealed. Should a player end the game
by taking a deal, a pseudo-game is continued from that point to see how much the player could have won by
remaining in the game. Depending on subsequent choices and offers, it is determined whether or not the
contestant made a "good deal", i.e. won more than if the game were allowed to continue.
Since the range of possible values is known at the start of each game, how much the banker offers at any given
point changes based on what values have been eliminated. To promote suspense and lengthen games, the
banker's offer is usually less than the expected value dictated by probability theory, particularly early in the
game. Generally, the offers early in the game are very low relative to the values still in play, but near the end of
the game approach (or even exceed) the average of the remaining values.
If you were to select a suitcase at random, what is the expected value?
Expected Value =    xP( X  x) 
 1 
 1 
 1 
 1 
 1  3418416.01
($.01)    ($1)    ($5)      ($750000)    ($1000000)   
 $131, 477.54
26
26
26
26
26
 
 
 
 
 26 
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