Download Means and Variances of Random Variables

CHAPTER 7
Section 7.2 – Means and Variances 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.
EXAMPLE 7.5 - THE TRI-STATE PICK 3
 See
example 7.5 on p.407
 Probability
distribution of X:
Payoff X:
Probability:

$0
0.999
$500
0.001
Mean of the random variable X is found by:
$500

1
999
+ $0
= $0.50
1000
1000
In other words, you are expected to lose $0.50 per ticket if many
tickets are purchased over time.
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.
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;
𝜇𝑋 = x1 p1 + x2 p2+ …….xk pk = x𝑖 p𝑖
MEAN AND EXPECTED VALUE
 The
mean of a probability distribution
describes the long-run average outcome.
 You
will often find the mean of a random
variable X called expected value of X.
 The
common symbol μ, the Greek letter mu,
is used to represent the mean of a
probability distribution (expected value).

Some other common notations include:
𝜇𝑋 (this is the most common)
𝜇 𝑋
 𝐸(𝑋)

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 that accompany the choice of
the mean to measure center.
Recall from chapter 2 that the variance of a data
set is written as 𝑠 2 and it represents an average of
the squared deviation from the mean.
To distinguish between the variance of a data set
and the variance of a random variable X, we write
the variance of a random variable X as
THE VARIANCE OF RANDOM VARIABLE
Definition:
Suppose that X is a discrete random variable whose probability
distribution is
Value:
x1 x2 x3 …
Probability: p1 p2 p3 …
and that µX is the mean of X. The variance of X is
Var(X)   X2  (x1   X ) 2 p1  (x 2   X ) 2 p2  (x 3   X ) 2 p3  ...
  (x i   X ) 2 pi
The standard deviation σx of X is the square root of the variance.
EXAMPLE 7.7 - SELLING AIRCRAFT PARTS

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
0.3
5000
0.4

Calculate the mean and variance of X

See p.411 to check your answers
10,000
0.2
STATISTICAL ESTIMATION AND THE
LAW OF LARGE NUMBERS
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
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To estimate μ, we choose a SRS of young women and
use the sample mean 𝑥 to estimate the unknown
population mean μ.
Statistics obtained from probability samples are
random variables because their values would vary in
repeated samplings.
It seems reasonable to use 𝑥 to estimate μ.
A SRS should fairly represent the population, so the
mean 𝑥 of the sample should be somewhere near the
mean μ of the population.
Of course, we don’t expect 𝑥 to be exactly equal to μ,
and realize that if we choose another SRS, the luck of
the draw will probably produce a different 𝑥.
LAW OF LARGE NUMBERS
If we keep on adding observations to our random
sample, the statistic 𝑥 is guaranteed to get as close
as we wish to the parameter μ and then stay that
close.
 This remarkable fact is called the law of large
numbers.
 The law of large numbers states the following:

Draw independent observations at random from any
population with finite mean μ.
 Decide how accurately you would like to estimate μ.
 As the number of observations drawn increases, the
mean 𝑥 of the observed values eventually approaches
the mean μ of the population as closely as you specified
and then stays that close.


See example 7.8 on p.414
THE “LAW OF SMALL NUMBERS”
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
Both the rules of probability and the law of large
numbers describe the regular behavior of chance
phenomena in the long run.
Psychologists have discovered most people
believe in an incorrect “law of small numbers”

That is, we expect even short sequences of random
events to show the kind of average behavior that in
fact appears only in the long run.
HOW LARGE IS A LARGE NUMBER?


The law of large numbers says that the actual
mean outcome of many trials gets close to the
distribution mean μ as more trials are made.
It doesn’t say how many trials are needed to
guarantee a mean outcome close to μ.

Homework:
p.412-417 #’s 24, 29, & 32
 Work on ch.6 & 7 extra credits
