Section 7.2 P1Means and Variances of Random Variables

AP Statistics

AP Statistics, Section 7.2, Part 1 2

Random Variables: Mean

1 1 2 2 3 3X n n

X i i

p x p x p x p x

p x

AP Statistics, Section 7.2, Part 1 3

Random Variables: Example

The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn.

What is the average winnings?

.001(500) .999 0

.50 0

.50

X

AP Statistics, Section 7.2, Part 1 4

Random Variables: Example

The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn.

What is the average PROFIT?

Mean = Expected Value

.001(499) .999 1

.499 .999.50

X

AP Statistics, Section 7.2, Part 1 5

Random Variables: Variance

2 2 221 1 2 2

22

X x x n n x

X i i x

p x p x p x

p x

AP Statistics, Section 7.2, Part 1 6

Random Variables: Example

The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn.

What is the average winnings?

2 2

2

(500 .5) .001

0 .5 .999

249.50025 .24975249.7515.8

X

AP Statistics, Section 7.2, Part 1 7

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-bar of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close.

AP Statistics, Section 7.2, Part 1 8

Example

The distribution of the heights of all young women is close to the normal distribution with mean 64.5 inches and standard deviation 2.5 inches.

What happens if you make larger and larger samples…

AP Statistics, Section 7.2, Part 1 9

AP Statistics, Section 7.2, Part 1 10

Law of Small Numbers

Most people incorrectly believe in the law of small numbers.

“Runs” of numbers, etc.

AP Statistics, Section 7.2, Part 1 11

AP Statistics, Section 7.2, Part 1 12

Rules for Means

Rule 1: The same scale change of elements of a probability distribution has the same effect on the means.

Rule 2: The mean of sum of the two distributions is equal to the sum of the means.

a bX X

X Y X Y

a b

AP Statistics, Section 7.2, Part 1 13

Rule 1 Example

A company believes that the sales of product X is as follows.

X 1000 3000 5000 10,000

P(X) .1 .3 .4 .2

1000 .1 3000 .3 5000 .4 10000 .25000 units

X

X

AP Statistics, Section 7.2, Part 1 14

Rule 1 Example

If the expected profit on each sale of Product X is $2000, what is the overall expected profit?

0 2000 0 2000 10,000,000X X

1000 .1 3000 .3 5000 .4 10000 .25000 units

X

X

AP Statistics, Section 7.2, Part 1 15

Rule 1 Example

A company believes that the sales of product Y is as follows.

Y 300 500 750

P(Y) .4 .5 .1

300 .4 500 .5 750 .1445 units

Y

Y

AP Statistics, Section 7.2, Part 1 16

Rule 1 Example

If the expected profit on each sale of Product Y is $3500, what is the overall expected profit?

0 3500 0 3500 1,557,500Y Y

300 .4 500 .5 750 .1445 units

Y

Y

AP Statistics, Section 7.2, Part 1 17

Rule 2 Example

What is the total expected profits combined of both Product X and Product Y?

2000 3500 10,000,000 1,575,50011,557,500

X Y

2000

3500

10,000,0001,557,500

X

Y

AP Statistics, Section 7.2, Part 1 18

Rules for Variances of Independent Distributions Only if the distributions are

independent can you apply these rules…

Rule 1: If a scale change involves a multiplier b, the variance changes by the square of b.

Rule 2: The variance of the sum of the two distributions is equal to the sum of the variances.

Rule 2b: The variance of difference of the two distributions is equal to the sum of the variances.

2 2 2

2 2 2

2 2 2

a bX X

X Y X Y

X Y X Y

b

AP Statistics, Section 7.2, Part 1 19

Example

The Daily 3 lottery has the following mean and variance for its payout:

What is the mean and variance of the winnings?

2

.50

249.7515.80

X

X

X

1

21

1

.50

249.7515.80

X

X

X

AP Statistics, Section 7.2, Part 1 20

Example

The Daily 3 lottery has the following mean and variance for its payout:

What is the mean and variance of the payouts of playing twice?

2

.50

249.7515.80

X

X

X

2

.50 .50 1.00

249.75 249.75

22.34

X X

X X

X X

AP Statistics, Section 7.2, Part 1 21

Example The Daily 3 lottery

has the following mean and variance for its payout:

What is the mean and variance of the payouts of playing every day of the year?

2

.50

249.7515.80

X

X

X

2

.50 365 182.5

249.75 365 91158.75

301.92

X X X

X X X

X X X