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Day 7 – Expectation
EXPECTED VALUE
There are three main measures that are used to describe the central tendency of a set of data :
mean, median and mode
For probability distribution functions, mean, median and mode of a random variable X are
defined as follows :
a) Mean or Expected Value ( E ( X ) )
i n
E ( X )   xi P ( X  xi )
i 1
E ( X )  x1P( X  x1 )  x2 P ( X  x2 ) 
 xn P ( X  x n )
We can think of the expected value of a probability distribution as the “weighted average” or as
the “long-run average”
b)  Median of X
 the value with half the probability below and half the probability above.
c)  Mode of X
 the most common value.
PROPERTIES OF EXPECTED VALUE

E(a) = a, where a is a constant,

E(aX) = aE(X), where a is a constant,

E  f  X     f  xi   P  X  xi  , where f(X) is some real valued function of the
i n
i 1
random variable X

E  aX  b   aE  X   b , where a and b are constants
Example 1) : Find the following below:
x
0
1
2
3
P( X  x)
1/6
1/2
1/5
2/15
a.
E( X )
b.
E( X 2 )
c.
E ( X 2  3 X  1)
Example 2) : A dart board has concentric circles of a radius of 1, 2, and 3. A dart is randomly
thrown and if it lands on the small circle a player gets $8.00, the middle region $6.00, the outer
region $4.00 and if it misses the board he or she loses $7.00. The probability of missing the
board is 0.5.
a.
How will the player fare in this game?
b.
What can you say about this game?
VARIANCE AND STANDARD DEVIATION
 The variance of a random variable is a measure of how spread out a random variable
is. The symbol for variance is Var ( X ) or
 2.
 Variance associated with a random variable X can be calculated as follows:
i n
Var ( X )  E (( X   ) )    x    P( X  x)
2
2
i 1
 Variance = average of the squared deviations about the mean
 Variance can also be calculated as follows:
Var ( X )  E ( X 2 )  ( E ( X ))2  E ( X 2 )   2
 The standard deviation of a random variable is also a measure of how spread out
random variable is
Standard deviation 
Sd  X    2  
PROPERTIES OF THE VARIANCE

Var(a) = 0.

Var(aX) = a2Var(X), where a is a constant

Var(aX + b)= a2Var(X), where a and b are constant