Download Random Variables - University of Arizona Math

Random Variables
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A Random Variable assigns a numerical value
to all possible outcomes of a random
experiment
We do not consider the actual events but we
associate numbers with the events that arise
from the experiment
Random Variables, con’t
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Consider the experiment of tossing a fair coin three
times
What is the sample space associated with this
experiment:
TTT,TTH,THT,HTT,HHT,HTH,THH,HHH
In this case, we may say that our random variable, X,
records the number of heads in three tosses
What are the possible values for X?
In this case, the random variable is used to describe
certain events.
Random Variables, con’t
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In general, given an experiment and a random
variable, X, and a number, x, the expression X=x
stands for the event that the random variable takes on
the value of x
X=0 is the event that no heads occur, TTT
X=1 is the event that exactly one head occurs
TTH,THT,HTT
X=2 is the event that exactly two heads occur
HHT,HTH,THH
X=3 is the event that exactly three head occurs HHH
Random Variables, con’t
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Since the number, X, that is associated with
outcomes takes on different values by chance,
it is called a random variable.
Random Variables, con’t
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TTT,TTH,THT,HTT,HHT,HTH,THH,HHH
We let P(X=x) be the probability that the event
X=x occurs
What is P(X=0)?
What is P(X=1)?
What is P(X=2)?
What is P(X=3)?
Random Variables, con’t
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Let’s make a table with
all possible values of x
and the corresponding
probabilities
Notice that the
probabilities sum to 1
Total
x
P(X=x)
0
1/8
1
3/8
2
3/8
3
1/8
8/8=1
Random Variables, con’t
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TTT,TTH,THT,HTT,HHT,HTH,THH,HHH
We may also have probabilities such as:
P(Xx), P(Xx), P(aXb), P(Xx)
What is P(0X1)?
What is P(0X<2)?
What is P(X3)?
What is P(X5)?
Expected Value of a Random
Variable
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Let’s play a game: We will roll a fair die. If we
roll a 1 or 2 you lose $5, if we roll a 3 or 4 you
don’t owe me anything, if we roll a 5 or 6 you
win $20.
Who wants to play one time?
Who wants to play three times?
Who wants to play ten times?
Expected Value of a Random
Variable
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The probability experiment is
tossing a die
Sample space 1, 2, 3, 4, 5,
6
Let’s think about the game:
Let X be the amount won on
one play of the game, what
values can X take on?
x
P(X=x)
Expected Value of a Random
Variable, con’t
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E(X) is the Expected value of a random variable X
E(X) is the average value that the random variable, X, would take
on after infinitely many trials
Suppose that X can assume only the distinct values x1, x2,..., xn.
The expected value of X, denoted by E(X), is the sum of these
values, weighted by their respective probabilities. That is,
n
E( X ) 
 xi  P  X  xi 
i 1
Expected Value of a Random
Variable, con’t
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Let’s look at the
expected value or
the average
amount that you
would win over the
long run
x
P(X=x)
xP(x=x)