Download 7.2 Means and Variances of Random Variables

7.2 Means and Variances of
Random Variables
Review
Review
Review
Review
Mean of a Discrete Random
Variable
The mean value of a discrete random
variable x, denoted by mx, is computed by
first multiplying each possible x value by the
probability of observing that value and then
adding the resulting quantities.
Symbolically,
m X   x  p(x)
all possible
values of x
Example
• A professor regularly gives multiple
choice quizzes with 5 questions. Over
time, he has found the distribution of the
number of wrong answers on his
quizzes is as follows
x
0
1
2
3
4
5
P(x)
0.25
0.35
0.20
0.15
0.04
0.01
Example
• Multiply each x value by its
probability and add the results to
get mx.
x
0
1
2
3
4
5
P(x)
0.25
0.35
0.20
0.15
0.04
0.01
x•P(x)
0.00
0.35
0.40
0.45
0.16
0.05
1.41
mx = 1.41
Variance and Standard Deviation of
a Discrete Random Variable
The Variance of a Discrete Random Variable x,
2
denoted by  x is computed by fist subtracting the
mean from each possible x value to obtain the
deviations, then squaring each deviation and
multiplying the result by the probability of the
corresponding x value, and then finally adding
these quantities. Symbolically,
2
 x  X
The standard deviation of x, denoted by x, is
the square root of the variance.
X2   (x  m )2  p(x)
all possible
values of x
Previous Example continued
x
0
1
2
3
4
5
P(x)
0.25
0.35
0.20
0.15
0.04
0.01
x•P(x)
0.00
0.35
0.40
0.45
0.16
0.05
1.41
x-m
-1.41
-0.41
0.59
1.59
2.59
3.59
(x - m (x - m•P(x)
1.9881
0.4970
0.1681
0.0588
0.3481
0.0696
2.5281
0.3792
6.7081
0.2683
12.8881
0.1289
1.4019
Variance    1.4019
2
X
Standard deviation
  x  1.4019  1.184
The Mean & Variance of a
Linear Function
If x is a random variable with mean mx
2
and variance X and a and b are
numerical constants, the random variable
y defined by y = a + bx is called a linear
function of the random variable x.
The mean of y = a + bx is my = ma + bx = a + bmx
The variance of y is  2y  a2bx  b2 X2
From which it follows that the standard deviation
of y is  y  abx  b  x
Example
Suppose x is the number of sales staff
needed on a given day. If the cost of doing
business on a day involves fixed costs of
$255 and the cost per sales person per day
is $110, find the mean cost (the mean of x or
mx) of doing business on a given day where
the distribution of x is given below.
x
1
2
3
4
p(x)
0.3
0.4
0.2
0.1
Example continued
We need to find the mean of y = 255 + 110x
x p(x) xp(x)
1 0.3
0.3
2 0.4
0.8
3 0.2
0.6
4 0.1
0.4
2.1
m  2.1
x
m y  m 255110 x  255  110m x
 255  110(2.1)  $486
Example continued
We need to find the variance and standard
deviation of y = 255 + 110x
x
1
2
3
4
2
p(x) (x-m) p(x)
0.3 0.3630
0.4 0.0040
0.2 0.1620
0.1 0.3610
0.8900

2
2
x
 
x
0.89  0.9434
 (110)   (110) (0.89)  10769
2
255 110 m X

  0.89
255 110 m X
2
x
2
 110 x  110(0.9434)  103.77
Means and Variances for
Linear Combinations
If x1, x2,  , xn are random variables and a1,
a2,  , an are numerical constants, the
random variable y defined as
y = a1x1 + a2x2 +  + anxn
is a linear combination of the xi’s.
Means and Variances for Linear
Combinations
If x1, x2,  , xn are random variables with means
m1, m2,
and

 , ,
, mn and variances
2
1
,
2
2
2
n
respectively,
y = a1x1 + a2x2 +  + anxn then
1. my = a1m1 + a2m2 +  + anmn
(This is true for any random variables with no conditions.)
2. If x1, x2,  , xn are independent random
variables then
2y  a1212  a2222 
 an2n2
and
 y  a1212  a2222 
 an2n2
Example
A distributor of fruit baskets is going to put 4
apples, 6 oranges and 2 bunches of grapes in
his small gift basket. The weights, in ounces, of
these items are the random variables x1, x2 and
x3 respectively with means and standard
deviations as given in the following table.
Apples Oranges Grapes
Mean
m
Standard deviation

8
10
7
0.9
1.1
2
Find the mean, variance and standard deviation of the
random variable y = weight of fruit in a small gift
basket.
Example continued
It is reasonable in this case to assume that the
weights of the different types of fruit are
Apples Oranges Grapes
independent.
Mean
m
Standard deviation

8
10
7
0.9
1.1
2
a1  4, a2  6, a3  2, m1  8, m 2  10, m 3  7
1  0.9, 2  1.1, 3  2
m y  ma x a x
1
1
2
2
 a3 x 3
 a1m1  a2m 2  a3m 3
 4(8)  6(10)  2(7)  106
2y  a2 x a x
1
1
2
2
 a3 x 3
 a12 12  a2222  a3232
 42 (.9)2  6 2 (1.1)2  22 (2)2  72.52
y = 72.52  8.5159
Summary
• The Law of Large Numbers says that the
average of the values of X observed in many
trials approach μ.
• If X is discrete with the possible values xi, the
means is the average of the values of X, each
weighted by its probability:
μX = x1p1 + x2p2 + … + xnpn
• The variance σ2 for a discrete variable:
σ2X = (x1 - μ)2p1 + (x2 – μ)2p2 + … + (xn - μ)2pn
• The standard deviation σX is the square root of
the variance.
Summary
The means and variance of random variables obey
the following rules
• If a nd b are fixed numbers, then
μa + bX = a + b μX
σ2a+bX = b2σ2X
• If X and Y are any two random variables, then
μX + Y = μX + μY
• And if X and Y are independent, then
σ2 X + Y = σ2 X + σ2 Y
σ2X – Y = σ2X + σ2Y