Means and Variances of Means and Variances of Random VariablesRandom Variables

Means and Variances of Means and Variances of Random VariablesRandom Variables

Activity 1: means of random VariablesActivity 1: means of random Variables

To see how means of random variables work, consider a random variable that takes values {1,1,2,3,5,8}. Then

do the following:

To see how means of random variables work, consider a random variable that takes values {1,1,2,3,5,8}. Then

do the following:1.Calculate the mean of the population:

2.Make a list of all the sample of size 2 from this population. You should have 15 subsets of size 2

3.Find the mean of the 15 x-bar in the third column and compare the result with the population mean.

4.Repeat steps 1-3 for a different (but still small) populations of your choice. Now compare your result with each other.

5.Write a brief statement that describes what you discovered.

1.Calculate the mean of the population:

2.Make a list of all the sample of size 2 from this population. You should have 15 subsets of size 2

3.Find the mean of the 15 x-bar in the third column and compare the result with the population mean.

4.Repeat steps 1-3 for a different (but still small) populations of your choice. Now compare your result with each other.

5.Write a brief statement that describes what you discovered.

Mean of a random VariableMean of a random Variable

The mean of a discrete

random variable X is a

weighted average of the

possible values that the

random variable can take.

The mean of a discrete

random variable X is a

weighted average of the

possible values that the

random variable can take.

exampleexampleA Tri-State Pick 3 game in New Hampshire,

Maine and Vermont let you choose three-digit

number and the state chooses three-digit

winning number at random and pays you $500 if

your number is chosen. Because there are 1000

three digit numbers, you have a probability of

1/1000 of winning. Taking X to be the amount

your ticket pays you, the probability distribution

of X is:

A Tri-State Pick 3 game in New Hampshire,

Maine and Vermont let you choose three-digit

number and the state chooses three-digit

winning number at random and pays you $500 if

your number is chosen. Because there are 1000

three digit numbers, you have a probability of

1/1000 of winning. Taking X to be the amount

your ticket pays you, the probability distribution

of X is:

What is the average payoff from the tickets?What is the average payoff from the tickets?

Payoff X $0 $500

Probability 0.999 0.001

($0 + $500)/2 = $250($0 + $500)/2 = $250

The long-run average pay off is: The long-run average pay off is:

$500(1/1000) + $0(999/1000)$500(1/1000) + $0(999/1000)= $0.50= $0.50

Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The common symbol for the mean (also known as the expected value of X) is , formally defined by

The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.

Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The common symbol for the mean (also known as the expected value of X) is , formally defined by

The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.

Benford’s LawBenford’s LawCalculating the expected first digitCalculating the expected first digit

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

First digit V

1 2 3 4 5 6 7 8 9

Probability

0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

Find the Find the of the X distributionof the X distribution

y= 3.4441y= 3.4441Find the Find the of the V distributionof the V distribution

x= 5x= 5

x=

5

x=

5

y=

3.4441

y=

3.4441

Variance of discrete Random VariablesVariance of discrete Random Variables

The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by:

The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by:

The standard deviation is the square root of the

variance.

The standard deviation is the square root of the

variance.

Gabby Sells Cars:Gabby Sells Cars:

Gabby is a sales associate at a large auto

dealership. She motivates herself by using

probability estimates of her sales. For sunny

Saturday in April, she estimates her car sales as

follows:

Gabby is a sales associate at a large auto

dealership. She motivates herself by using

probability estimates of her sales. For 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

Let’s find the mean and variance of XLet’s find the mean and variance of X

Cars sold: X 0 1 2 3

Probability: P 0.3 0.4 0.2 0.1

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

1.1 0.890

Law of Large numberLaw of Large number

Rule for MeansRule for Means

Rules for VariancesRules for Variances

Example:Example:

Cars sold: X 0 1 2 3

Probability: P 0.3 0.4 0.2 0.1

Trucks and SUV: Y 0 1 2

Probability: P 0.4 0.5 0.1

Mean of X:

1.1 cars

Mean of X:

1.1 carsMean of Y:

0.7 T’s & SUV’s

Mean of Y:

0.7 T’s & SUV’s

At her commission rate of 25% of gross profit on each vehicle she sells, Linda

expects to earn $350 for each cars sold and $400 for each truck and SUV’s

sold. So her earnings are:

At her commission rate of 25% of gross profit on each vehicle she sells, Linda

expects to earn $350 for each cars sold and $400 for each truck and SUV’s

sold. So her earnings are:

Z= 350 X + 400YZ= 350 X + 400YCombining rule 1 and 2 her mean earnings

will be:

Combining rule 1 and 2 her mean earnings

will be:Uz= 350 UX + 400 UYUz= 350 UX + 400 UY

Uz= 350 (1.1) + 400 (.7)Uz= 350 (1.1) + 400 (.7)= $665 a day= $665 a day