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Expectation
Random Variables
Graphs and Histograms
Expected Value
Random Variables
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A random variable is a rule that assigns a numerical
value to each outcome of an experiment. We will
classify random variables as either
Finite discrete – if it can take on only finitely many
possible values.
Infinite discrete – infinitely many values that can be
arranged in a sequence.
Continuous – if its possible values form an entire
interval of numbers
Example One
Suppose that we toss a fair coin three times. Let
the (finite discrete) random variable X denote the
number of heads that occur in three tosses. Then
Sample
Pt.
Value of X
Sample
Pt.
Value of X
HHH
HHT
HTH
3
2
2
HTT
THT
TTH
1
1
1
THH
2
TTT
0
Example Two
Suppose that we toss a coin repeatedly until a head
occurs. Let the (infinite discrete) random variable Y
denote the number of trials.
Sample
Pt.
Value of Y
Sample
Pt.
Value of Y
H
TH
TTH
1
2
3
TTTTH
TTTTTH
TTTTTTH
5
6
7
TTTH
4


Example Three
A biologist records the length of life (in hours)
of a fruit fly. Let the (continuous) random
variable Z denote the number of hours
recorded. If we assume for simplicity, that
time can be recorded with perfect accuracy,
then the value of Z can take on any
nonnegative real number.
Graphs and Histograms
Given a random variable X, we will be
interested in the probability that X takes on a
particular real value x, symbolically we write
pX(x) = P(X = x)
pX(x) is referred to as the probability function of
the random variable X.
Geometric Representation
Consider Example Two where a coin is tossed
three times. From the given table we see that
Histogram
p(0)= P(X= 0)= 1/8
p(1)= P(X= 1)= 3/8
p(2)= P(X= 2)= 3/8
p(3)= P(X= 3)= 1/8
P(x)-values
4/8
3/8
2/8
1/8
0
0
1
2
x-values
3
Line and Bar Graphs
Bar Graph
Line Graph
4/8
p(x)-values
p(x)-values
4/8
3/8
2/8
`
1/8
3/8
2/8
`
1/8
0
0
0
1
2
x-values
4
0
1
2
x-values
4
Expectation
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x1  x2    xn
Arithmetic Mean x 
n
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Consider 10 hypothetical test scores:
65, 90, 70, 65, 70, 90, 80, 65, 90, 90

Calculate the mean as follows:
3  65  2  70  1 80  4  90 3
2
1
4
x
 65  70  80  90  77.5
10
10
10
10
10
Expectation
We may express the arithmetic mean as:
fk
f1
f2
x  x1  x2    xk
n
n
n
As the number of repetition increases
fi
xi  pi
n