Download Section 7.2 Part 1 – Means and Variances of Random Variables

Section 7.2 Part 1 – 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.
o
See example 7.5 on p.407
Probability distribution of X:
Payoff X:
$0
Probability:
$500
0.999
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 ____________________________is their _______________________.

The mean of a_________________________is also a ___________________________________________, but
with an essential change to take into account the fact that __________________need to be ________________.
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 _____, the Greek letter mu, is used to represent the mean of a probability distribution
(expected value).

Some other common notations include:

𝜇𝑋

𝜇(𝑋)

𝐸(𝑋)
(this is the most common)
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 _____________________________ the choice of
_______________ to measure center.

Recall from chapter 2 that the variance of a data set is written as ___________ 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
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
The standard deviation ______ of X is the _____________________________
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:
1000
Probability:
3000
0.1
0.3
5000
0.4
10,000
0.2
Calculate the mean and variance of X
See p.411 to check your answers
Statistical Estimation and the Law of Large Numbers

To estimate μ, we choose a SRS of young women and _____________________________________the unknown
________________________________.

Statistics obtained from ___________________________are __________________________because their
values would ___________________________________.

It seems reasonable to use ______ to estimate ______.

A SRS should fairly represent the ________________, so the mean 𝑥̅ of the sample should be
________________________________ μ of the population.

Of course, we don’t expect 𝑥̅ to be exactly equal to μ, and realize that if we choose another SRS, the luck of the
draw will probably produce a different 𝑥̅ .
Law of Large Numbers

If we keep on adding observations to our random sample, the statistic 𝑥̅ is guaranteed to get
_______________________to the _____________________and then _____________________.

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 ______________ you would like to estimate μ.

As the number of observations draw _________________, the mean 𝑥̅ of the observed values eventually
_________________________ of the population ________________________and then stays that close.

See example 7.8 on p.414
The “Law of Small Numbers”

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 __________________ “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.