7.2 Means and Variances of Random Variables. Calculate the mean and standard deviation of random...

AP Statistics7.2 Means and Variances of Random

Variables

Calculate the mean and standard deviation of random variables

Understand the law of large numbers

Learning Objective:

You pick a three digit number in the lottery. If your number matches the states number, you win $500. What are your average winnings?

Probability Distribution

  Probabilities are an idealized description of long-

run proportions, so the mean of a probability distribution describes the average winnings in the long-run

The mean of a Random Variable

Outcome $500 $0

Probability 1/1000 999/1000

Suppose X is a discrete random variable whose distribution is

To find the mean of X, multiply each possible value by its probability, then add all the products:

Mean of a Discrete Random Variable

Value of X x₁ x₂ x₃ ....

Probability p₁ p₂ p₃ ....

Ex: The distribution of the count X of heads in four tosses of a balanced coin.

The expected value is: (0)(0.0625) + (1)(0.25)+….+(4)

(0.0625)= 2

# of heads

0 1 2 3 4

Prob. 0.0625 0.25 0.375 0.25 0.0625

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

# of heads

# of heads

Construct a histogram of the distribution.

The mean is the center of a symmetrical distribution.

What do you notice about the mean?

μ= 1(0.25)+2(0.32)+…+7(0.01)= 2.6

Ex: What is the mean number of inhabitants in an American

household?Inhabitants

1 2 3 4 5 6 7

Probability 0.25 0.32 0.17 0.15 0.07 0.03 0.01

7.17 (0)(0.1)+(1)(0.15)+…+(4)(0.15)=2.25

7.18Your payout is either $0 or $3

a)

b) 0(0.75)+3(0.25)=$0.75

c) The casino makes $0.25 for every dollar you bet

Complete 7.17-7.19 (pg.389)

Payout $0 $3

Probability 0.75 0.25

7.19If you choose a #, you could get:

abc, acb, bac, bca, cab, cbaEx: 345, 354, 435, 453, 534, 543

0(0.994)+83.33(0.006)=$0.50

Payout $0 $83.88

Probability 0.994 0.006

Law of large numbers Draw independent observations at random from any

population with finite mean (μ). Decide how accurately you would like to estimate the

mean. As the number of observations drawn increases, the mean of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close.

  Describe this in your own words?

When you increase your sample size, your sample mean gets closer to the true mean

Statistical estimation and the law of large numbers

Reese’s example: http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html

Ex: Use the average height of women to explain this. The mean is 64.5 in with a standard deviation of 2.5 in.

Rule 1: If X is a random variable and a and b are fixed numbers, then:

Rule 2: If X and Y are (independent) random variables, then

Rules for Means

Ex: Military divisions

Civilian Division 

Let x= # of military units sold y= # of civilian units sold What is the mean number of military units

sold? Civilian units sold?µ=1000(.1)+3000(.3)+…+10000(.2)=5000 units

µ=300(.4)+500(.5)+750(.1)= 445 units

Units 1000 3000 5000 10000

Prob 0.1 0.3 0.4 0.2

Units 300 500 750

Prob. 0.4 0.5 0.1

If a profit of $2000 is made on each military unit sold and $3500 is made on each civilian unit, what is the total mean profit for units sold?

The variance of X is:

So the standard deviation is:

Variance of a Discrete Random Variable

Value of X x₁ x₂ x₃ ....

Probability p₁ p₂ p₃ ....

Then sum(L₃)

Steps in the calculator:

Find the standard deviation of the military units sold?

Find the standard deviation of the civilian units sold?

Rule 1:

Rule 2:

***we can’t add standard deviations, only variances!!!!***

Rules for Variance

Ex: The payoff X of a $1 ticket in the Tri-State pick 3 game is $500 with probability 1/1000 and $0 the rest of the time. What is the variance of the total payoff if you buy $1 ticket on two different days?

x $0 $500

P(x) 0.999 0.001

SAT math score X SAT verbal score Y What are the mean and standard deviation

of the total score X + Y among students applying to this college?

µ=625+590=1215

σ= √(90²+100²)=134.54

SAT scores

Tom’s score X:

George’s score Y:

Their scores vary independently. What is the mean difference between their scores?

 110-100=10

What is the variance of the difference between their scores?  10²+8²=164

So the standard deviation is? √164=12.8

Golf scores

What is the mean if I doubled everyone’s test score?

2(80)=160

What is the standard deviation? √ (2²*4²)=8

The class average on the last chapter test was 80 with a

standard deviation of 4.

What if I added 5 bonus points to everyone’s score. What is the new mean and standard deviation?

μ=80+5=85σ=4 What if I doubled everyone’s score and

added 5 points. What is the new mean and standard deviation?

μ=2(80)+5=165σ= √ (2²*4²)=8