A. P. STATISTICS
LESSON 7.2
MEANS AND VARIANCES OF RANDOM VARIABLES
ESSENTIAL QUESTION:
How are means and variances
found for random variables?
• To find the mean of random variables.
• To find the variance of random variables.
Means and Variances of
Random Variables
Probability is the mathematical language
that describes the long-run regular behavior
of random phenomena.
The probability distribution of a random
variable is an idealized relative frequency
distribution.
The Mean of a Random Variable
The mean x of a set of observations is their
ordinary average. The mean of a random
variable X is also an average of the possible
values of X, but with an essential change to
take into account the fact that not all
outcomes need to be equally likely.
Example 7.5
The Tri-State Pick 3
Page 407
Most states have lotteries where you choose
a three digit number. If a state charges a
dollar to play and $500 if you win, what is
your average payoff from many tickets?
Payoff X:
Probability:
$0
0.999
$500
0.001
$500 _ 1__ + $0 _999_ = $0.50
1000
1000
Mean and Expected Value
Just as probabilities are an idealized
description of long-run proportions, the mean
of a probability distribution describes the
long-run average outcome.
The common symbol μ, the Greek letter mu.
You will often find the mean of a random
variable X called expected value of X.
Mean 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
To find the mean of X, multiply each possible
value by its probability, then add all products;
μ = x1 p1 + x2 p2+ …….xk pk =∑xi pi
Example 7.6
Benford’s Law*
Page 408 - 409
* It is a striking fact that the first digits of number in legitimate records often
follow a a distribution known as Benford’s Law page 345
The Variance of
Random Variable
The mean is a measure of the center of a
distribution.
The variance and the standard deviation
are the measures of spread of that
accompany the mean to measure center.
We write the variance of a random
variable X as σ2.
Variance of a Discrete Random
Variable
Suppose that X is a discrete random variable
whose distribution is
Value of X: x1
Probability: p1
x2
p2
x3
p3
…
…
xk
pk
and that μ is the mean of X. The variance of X is
σ2x = (x1 – μx )2p1 + ( x2 – μx)2p + …+ ( xk – μx)2pk
= ∑ ( xi – μx )2 pi
The standard deviation σx of X is the square
root of the variance.
Example 7.7
Selling Aircraft Parts
Page 411
Gain Communications sells aircraft
communications units to both the military and the
civilian markets. Next year’s sales depend on
market conditions that cannot be predicted
exactly. Gain follows the modern practice of using
probability estimates of sales. The military division
estimates its sales as follows:
Units sold
Probability:
1000
0.1
3000 5000
0.3 0.4
10,000
0.2