Transcript Means and Variances of Random Variables

Mean of a Random Variable
Suppose that X is a discrete random variable whose
distribution is as follows :
Value of X
x1
x2
x3
...
xk
Probability
p1
p2
p3
...
pk
To find the mean of X, multiply each possible value by
its probability, then add all the products.
X = x1p1 + x2p2 + x3p3 + … + xkpk
=  xipi
Mean of a Random Variable
Example : Nelson is trying to get Homer to play a game
of chance. This game costs $2 to play. The game is to
flip a coin three times, and for each head that appears,
Homer will win $1.
Here is the probability distribution :
Number of Heads
Probability
0
1
2
3
.125
.375
.375
.125
What is the expected payoff of this game?
The mean!
Mean of a Random Variable
Number of Heads
Probability
0
1
2
3
.125
.375
.375
.125
X = (0)(.125) + (1)(.375) + (2)(.375) + (3)(.125) = 1.5
In this game, Homer would expect to win $1.50
So, should Homer play this game ? No.
Rules for Means
1) If X is a random variable and a and b are fixed numbers,
then :
 a+bX = a + bX
Example : What if Homer tries Nelsons game, but he
decides to award $10 for every head that Nelson flips.
Number of Heads
Probability
0
10
20
30
.125
.375
.375
.125
X = (0)(.125) + (10)(.375) + (20)(.375) + (30)(.125) = 15
Rules for Means
1) If X is a random variable and a and b are fixed numbers,
then :
 a+bX = a + bX
Example : What if Homer tries Nelsons game, but he
decides to award $10 for every head that Nelson flips.
Number of Heads
Probability
0
10
20
30
.125
.375
.375
.125
Notice that this is just a linear transformation with
b = 10 and a = 0.
10X = a + bX = 0 + 10X =
(10)(1.5) = 15
(Same as before)
Rules for Means
2) If X and Y are random variables, then
X+Y = X + Y
Example :
Let X be a random variable with X = 12
Let Y be a random variable with Y = 18
What is the mean of X + Y ?
X+Y = X + Y = 12 + 18 = 30
Variance of a Discrete Random Variable
Suppose that X is a discrete random variable whose
distribution is :
Value of X
x1
x2
x3
...
xk
Probability
p1
p2
p3
...
pk
and that X is the mean of X. The variance of X is :
X2
= (x1 - X)2 p1 + (x2 - X)2 p2 + ... + (xk - X)2 pk
=  (xi - X)2 pi
The standard deviation
the variance.
X of X is the square root of
Variance of a Discrete Random Variable
Example : Find the variance of Nelson’s game.
Number of Heads
Probability
0
1
2
3
.125
.375
.375
.125
Recall the mean was 1.5
X2
= (x1 - X)2 p1 + (x2 - X)2 p2 + ... + (xk - X)2 pk
= (0 - 1.5)2 (.125) + (1 - 1.5)2 (.375) + (2 - 1.5)2 (.375) +
(3 - 1.5)2 (.125)
= .28125 + .09375 + .09375 + .28125 = 0.75
X =
0.75
= .8660254
Rules for Variances
1) If X is a random variable and a and b are fixed
numbers, then :
 a2+ bX
= b2  2
X
2) If X and Y are independent random variables, then
X2+ Y
=
 X2
+
 Y2
X2- Y
=
 X2
+
 Y2
Rules for Variances
Example : Mike’s golf score varies from round to round.
X = 88
X = 8
Tiger’s golf score vary from round to round.
Y = 92
Y = 9
Q: What is the expected difference between our
two rounds ?
X-Y
=
X -  Y
= 88 - 92 = -4
Rules for Variances
Example : Mike’s golf score varies from round to round.
X = 88
X = 8
Tiger’s golf score vary from round to round.
Y = 92
Y = 9
Q: What is the variance of the differences ?
X2- Y
=
 X2
+
 Y2
= 64 + 81 = 145
X-Y =
145
= 12.041594
The Law of Large Numbers
• Draw independent observations at random from any
population with finite mean 
• Decide how accurately you want to estimate 
• As the number of observations increases, the mean
of the observed values eventually approaches the
mean of the population.
Example:
M & M’s
• Brown 30%
• Red 20%
• Yellow 20%
• Blue 10%
• Green 10%
• Orange 10%
M & M’s
• Brown 30%
• Red 20%
• Yellow 20%
• Blue 10%
• Green 10%
• Orange 10%
Law of Large Numbers
During an experiment, as the number of experiments
grows larger, then the observed mean will grow closer
and closer to the actual mean.
M & M’s
• Brown 30%
• Red 20%
• Yellow 20%
Red
Totals
0
Brown
0
Yellow Blue
0
0
Green Orange
0
0
• Blue 10%
• Green 10%
• Orange 10%
Totals
Red
Brown
Yellow
Blue
Green
Orange
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Homework
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