Transcript 7.2.1 - GEOCITIES.ws

The Mean of a Discrete
Random Variable
Lesson 7.2.1
Starter 7.2.1
• An unfair six-sided die comes up 1 on half
of its rolls. The other half of its rolls are
evenly spread through the other 5
outcomes.
• If X is the number that comes up, write
PDF for X.
Objectives
• Evaluate the mean of a discrete random variable
from its PDF
• Use the other names and notations for the
mean:
– µx
– Expected Value
– E(x)
• Define what is meant by a “fair game”
California Standard 5.0
Students know the definition of the mean of a discrete
random variable and can determine the mean for a
particular discrete random variable.
The “Expected Value” of a discrete Random Variable
• Suppose you roll a fair die 600 times.
• How many times would you expect that each face comes
up?
• How many TOTAL SPOTS will you see in the 600 rolls?
• What is the average number of spots per roll?
• This is the expected value of X, also called E(x), where X
is the number of spots that shows on each roll
– Note that we do not “expect” to get this result on any one roll (it’s
actually impossible in this case)
– The expected value is the average result over the long run
• This is also called the MEAN of the discrete random
variable.
– The mean is denoted by the Greek letter µ or µx
The Formula for µ
• To find the mean (or expected value) of
any discrete random variable, multiply
each possible outcome by its probability
– µx = x1p1 + x2p2 + … + xipi
– µx = Σxipi
• Apply the formula to a fair die: What is the
mean of X when X is defined as the
number of spots that show when the die is
rolled?
• What is the mean of X in the starter?
Example
• Four coins are tossed and X is the number
of heads that show.
– Recall the PDF of X (it’s already in your notes)
– Find µx by using the formula
• µx = (0)(1/16) + (1)(4/16) + (2)(6/16) +
(3)(4/16) + (4)(1/16) = 2
Example
• A certain lottery is played by paying $1 for
a chance at a $500 prize. One of 1000
numbered balls is drawn. If your number
matches the ball, you win.
– Let X be the amount you are paid (ignore the
$1)
X
500
0
– Write the PDF of X
P(X)
.001
.999
• Calculate E(x)
– E(x) = (500)(.001) + (0)(.999) = .50
– So you expect to be paid $.50 on average for
every $1 game!
Now let’s take the $1 into account
• A certain lottery is played by paying $1 for a
chance at a $500 prize. One of 1000 numbered
balls is drawn. If your number matches the ball,
you win.
– Let X be the net amount you win
– Write the PDF of X
X
499
• Calculate E(x)
P(X)
.001
-1
.999
– E(x) = (499)(.001) + (-1)(.999) = -.50
– So you expect to lose $.50 on average for every
game!
• What should E(x) be to make the game fair?
– By definition, a game is fair if E(X) = 0
The Purpose of Odds
• You roll a fair die and bet $1 that a 4 comes up.
– What is the probability that you win the bet?
– What are the odds against winning?
– How much should you and your opponent bet?
• Assuming proper odds are used, let X be your
net winnings. Write the PDF and find E(X)
X
+5
-1
P(X)
1/6
5/6
• E(X) = (5)(1/6) + (-1)(5/6) = 0
Objectives
• Evaluate the mean of a discrete random variable
from its PDF
• Use the other names and notations for the
mean:
– µx
– Expected Value
– E(x)
• Define what is meant by a “fair game”
California Standard 5.0
Students know the definition of the mean of a discrete
random variable and can determine the mean for a
particular discrete random variable.
Homework
• Read pages 385 – 394
• Do problems 17 – 19, 21, 22, 23