The Variance of a Random Variable

1
The Variance of a Random Variable

• Lecture 35
• Section 7.5.1
• Fri, Mar 26, 2004

2
The Variance of a Discrete Random Variable

• Variance of a Discrete Random Variable The
  square of the standard deviation of that random
  variable.
• The variance of X is denoted by
• ?2 or Var(X)
• The standard deviation of X is denoted by ?.

3
The Variance and Expected Values

• The variance is the expected value of the squared
  deviations.
• That agrees with the earlier notion of the
  average squared deviation.
• Therefore,
• Var(X) E((X µ)2).

4
Example of the Variance

• Again, let X be the number of children in a
  household.

x P(X x)
0 0.10
1 0.30
2 0.40
3 0.20
5
Example of the Variance

• Subtract the mean (1.70) from each value of X to
  get the deviations.

x P(X x) x µ
0 0.10 -1.7
1 0.30 -0.7
2 0.40 0.3
3 0.20 1.3
6
Example of the Variance

• Square the deviations.

x P(X x) x µ (x µ)2
0 0.10 -1.7 2.89
1 0.30 -0.7 0.49
2 0.40 0.3 0.09
3 0.20 1.3 1.69
7
Example of the Variance

• Multiply each squared deviation by its
  probability.

x P(X x) x µ (x µ)2 (x µ)2?P(X x)
0 0.10 -1.7 2.89 0.289
1 0.30 -0.7 0.49 0.147
2 0.40 0.3 0.09 0.036
3 0.20 1.3 1.69 0.338
8
Example of the Variance

• Add up the products to get the variance.

x P(X x) x µ (x µ)2 (x µ)2?P(X x)
0 0.10 -1.7 2.89 0.289
1 0.30 -0.7 0.49 0.147
2 0.40 0.3 0.09 0.036
3 0.20 1.3 1.69 0.338
0.810 ?2
9
Example of the Variance

• Add up the products to get the variance.

x P(X x) x µ (x µ)2 (x µ)2?P(X x)
0 0.10 -1.7 2.89 0.289
1 0.30 -0.7 0.49 0.147
2 0.40 0.3 0.09 0.036
3 0.20 1.3 1.69 0.338
0.810 ?2 0.9 ?
10
Exercise

• Lets Return To Lets Do It! 7.23, p. 430.
• The variance of Profit Indoors is
• Var(Profit Indoors) 29.0
• Compute the variance of Profit Outdoors.
• Which setting (indoors or outdoors) exhibits the
  greater variability?

11
Alternate Formula for the Variance

• It turns out that
• Var(X) E(X2) (E(X))2.
• That is, the variance of X is the expected value
  of the square of X minus the square of the
  expected value of X.
• Of course, we could write this as
• Var(X) E(X2) µ2.

12
Example of the Variance

• One more time, let X be the number of children in
  a household.

x P(X x)
0 0.10
1 0.30
2 0.40
3 0.20
13
Example of the Variance

• Square each value of X.

x P(X x) x2
0 0.10 0
1 0.30 1
2 0.40 4
3 0.20 9
14
Example of the Variance

• Multiply each squared X by its probability.

x P(X x) x2 x2?P(X x)
0 0.10 0 0.00
1 0.30 1 0.30
2 0.40 4 1.60
3 0.20 9 1.80
15
Example of the Variance

• Add up the products to get E(X2).

x P(X x) x2 x2?P(X x)
0 0.10 0 0.00
1 0.30 1 0.30
2 0.40 4 1.60
3 0.20 9 1.80
3.70 E(X2)
16
Example of the Variance

• Then use E(X2) to compute the variance.
• Var(X) E(X2) µ2
• 3.70 (1.7)2
• 3.70 2.89
• 0.81.
• It follows that ? ?0.81 0.9.

17
Exercise

• Return once more to Lets Do It! 7.23, p. 430.
• Use the alternate formula to compute the variance
  of Profit Indoors.

18
Means and Standard Deviations on the TI-83

• Store the list of values of X in L1.
• Store the list of probabilities of X in L2.
• Select STAT gt CALC gt 1-Var Stats.
• Press ENTER.
• Enter L1, L2.
• Press ENTER.
• The list of statistics includes the mean and
  standard deviation of X.

19
Means and Standard Deviations on the TI-83

• Let L1 0, 1, 2, 3.
• Let L2 0.1, 0.3, 0.4, 0.2.
• Compute the statistics.
• Compute µ and ? for the Indoor and Outdoor
  distributions in Lets Do It! 7.23, p. 430.

20
Assignment

• Page 442 Exercise 56.
• Page 451 Exercises 91, 93, 94, 95, 96.
• Find the variance and standard deviation.