7.2 Day 1: Mean & Variance of

Random Variables

Law of Large Numbers

The Mean of a Random Variable

The mean x of a set of observations is their ordinary average, but how do you find the mean of a discrete random variable whose outcomes are not equally likely?

The Mean of a Random

Variable is known as its

expected value.

Ex 1: The Tri-State Pick 3

In the Tri-State Pick 3 game that New Hampshire shares with Maine and Vermont, you choose a 3-digit number and the state chooses a 3-digit winning number at random and pays you $500 if your number is chosen.

Since there are 1000 possible 3 digit numbers, your

probability of winning is 1/1000.

The probability distribution of X (the amount your ticket pays you)

Payoff X: $0 $500

Probability: 0.999 0.001

The ordinary average of the two possible outcomes is $250, but that makes no sense as the average because $0 is far more likely than $500.

In the long run, you would only receive $500 once in every

1,000 tickets and $0 in the remaining 999 of

the tickets

So what is the mean?

The long-run average payoff or mean for this random variable X is fifty cents.

This is also known as the Expected Value.

1 999$500 $0 $0.50

1000 1000

We will say that

μx = $0.50.

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 vlaue by its probability, then add all the products

μx = x1p1 + x2p2 + … + xkpk

= Σxipi

We will use μx to signify that this is

the mean of a random variable and not of a data

set.

Ex 2: Benford’s LawCalculating the expected first digit

What is the expected value of the first digit if each digit is equally likely?

μx = 1(1/9) + 2(1/9) + 3(1/9) + 4(1/9) + 5(1/9) + 6(1/9) + 7(1/9) + 8(1/9) + 9(1/9)

= 5

First Digit X 1 2 3 4 5 6 7 8 9

Probability 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

The expected value is μx = 5.

What is the expected value if the data obeys Benford’s Law?

μx = 1(.301) + 2(.176) + 3(.125) + 4(.097) + 5(.079) + 6(.067) + 7(.058) + 8(.051) + 9(.046)

= 3.441

First Digit X 1 2 3 4 5 6 7 8 9

Probability .301 .176 .125 .097 .079 .067 .058 .051 .046The expected

value is μx = 3.441.

Probability Histogram for equally likely outcomes 1 to 9

In this uniform distribution, the

mean 5 is located at the

center.

Probability Histogram for Benford’s Law

The mean is 3.441 in this right skewed distribution.

Recall…

Computing a measure of spread is an important part of describing a distribution (SOCS)

The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center.

Variance 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

And that the mean μ is the mean of X. The variance of X is

σx2 = (x1 – μx)2p1 + (x2 – μx)2p2 + … + (xk – μx)2pk

The standard deviation σx of X is the square root of the variance.

We will use σx2 to

signify the variance and σx for the

standard deviation.

Ex 3: Linda Sells Cars

Linda is a sales associate at a large auto dealership. She motivates herself by using probability estimates of her sales. For a sunny Saturday in April, she estimates her car sales as follows:

Cars Sold: 0 1 2 3

Probability: 0.3 0.4 0.2 0.1

Find the mean and variance.

μx = 1.1 σx2 = 0.890

xi pi xipi (xi – μx)2pi

0 0.3 0.0 (0 – 1.1)2(0.3) = 0.363

1 0.4 0.4 (1 – 1.1)2(0.4) = 0.004

2 0.2 0.4 (2 – 1.1)2(0.2) = 0.162

3 0.1 0.3 (3 – 1.1)2(0.1) = 0.361

The standard

deviation is σx = 0.943

The Law of Large Numbers

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.

Ex 4: Heights of Young Women(Law of Large Numbers)

The average height of young women is 64.5

in.

The Law of Small Numbers

The law of small numbers does not exist, although psychologists have found that most people believe in the law of small numbers.

Most people believe that in the short run, general rules of probability with be consistent.

This is a misconception because the general rules of probability only exist over the long run.

In the short run, events can only be characterized as random.

How large is a large number?

The law of large numbers does not state how many trials are necessary to obtain a mean outcome that is close to μ.

The number of trials depends on the variability of the random outcomes.

The more variable the outcomes, the more trials that are needed to ensure that the mean outcome x is close the distribution mean μ.