Chapter 7: Random Variables

“Horse sense is what keeps horses from betting on what people do.”

Damon Runyon

7.1 Discrete and Continuous Random Variables (pp 367-

380)When the outcomes of an event that produces random results are numerical, the numbers obtained are called random variables.The sample space for the event is just a list containing all possible values of the random variable.Section 7.1 introduces the concept of a random variable and the probabilities associated with the various values of the variable.

7.1 - ContinuedRandom variable: the outcome of a random phenomenonDiscrete random variables:

Have a countable number of possible valuesExample:

Flip a coin 4 timesNumber of heads obtained: 0, 1, 2, 3, 4Number of heads possible is a discrete random variable, x

TTHHTHHT

THHH THTH TTTHHTHH HTHT TTHTHHTH HTTH THTT

HHHH HHHT HHTT HTTT TTTTx 4 3 2 1 0Prob(x) 1/16 4/16 6/16 4/16 1/16

7.1 - ContinuedContinuous random variable

Takes all values in an interval of numbersHas a density curve associated with itExample:

x is a random number in the interval and is therefore a continuous random variable

RAND function generates values of x in the interval

Random numbers generated on TI83+ are rounded to 10 decimal places (so you are really looking at discrete!)

Distinction between > and can be ignored.

0 1,

0 1,

1Prob 0 5 0, Prob 0 5

2. .x x

7.1 - ContinuedVery common types of continuous random variables are represented in normal probability distributions

Random observations from a normal distribution can be distributed with a TI83+

randNorm generates 100 random numbers from a normal distribution with

and stores them into List1 SortA(L1) sorts list of random numbers in ascending order

150 4 100, , L

mean 50 and standard deviation 4

7.2 – Means and Variances of Random Variables (pp 385-404)

If x is a discrete random variable with possible values having probabilities then

ix

ip

1 1 2 2 of valuesmean ... k k

i i

x x p x p x p

sum x p

2

2 2 2

1 1 2 2

2

and of valvariance ues

... k k

i i

x

x p x p x p

sum x p

Example: A random variable x assumes the values 1, 2, 3 with respective probabilities 60%, 30%, and 10%

L1 L2 L3 L4 L5 L6

1

2

3

SUMS 1

ix ip i ix p ix 2

ix 2

i ix p

2

Law of Large NumbersThe actual mean of many trials gets close to the distribution mean as more trials are made.

Example: A coin is flipped numerous timesExpectation: 50% of the time you’ll get a head

10 flips --many times– HIGHLY LIKELY that in some of the trials you will have 30% or less HEADS100 flips –many times-- HIGHLY UNLIKELY that any trials will yield a percentage of HEADS that is 30% or less

Try on the calculator: binomcdf(10, 0.5, 3) and binomcdf(100, 0.5, 30)

These give the probabilities of flipping 10 coins with 3 or less heads and flipping 100 coins with 30 of less heads

Rule #1 for Means

7 5

Let 1 3 . The mean of is 2.

Now multiply each element by 7, then add 5.

12 26

And the mean 19 7 2 5

This illustrates Rule #1 from page 396.

RULE #1: If is a random variable and and

are fix

,

,

X x

X

S S

S

X a b

ed numbers a bX Xa b

Rule #2 for Means

1 3 mean 2

Let 5 9 22 with mean 12

1 5 1 9 1 22 3 5 3 9 3 22

6 8 10 12 23 25

The mean of 14 2 12

mean of mean mean

This illustrates Rule #2 on page 396 ---

,

, ,

, , , , ,

, , , , ,

( )

X

Y

X Y

X Y

X Y

X Y X Y

X Y X

S

S

S

S

S

S S S

Y

Rule #1 for Variances

3 5

23 5

Let 10 14 . The mean 12,

the standard deviation 2, and the variance 4

Multiply each element by 3 and then add 5.

35 47 . The mean 41, the s.d. 6

and the variance 36

N.B. 36 3 4 var 3

,

.

,

.

X

X

X

T

T

T

2

3 5

2 2 2

var

StDev 3StDev

This illustrates Rule #1 from page 400: If is a random

variable and and are fixed numbers, then

X

X X

a bX X

T

T T

X

a b

b

Rule #2 for Variances

ind

Let

epe

3 6 9 . Mean 9, s.d. 2 4494897,

variance 6

Assume that and are .

10 6 10 9 10 12 14 6 14 9 14 12

2 1 2 4 5 8

Mean 3, s.d. 3 16227766, var

nden

e

t

ianc 10

-

-

, , .

.

, , , , ,

, , , , ,

.

Y

X Y

X Y

X Y

T

T T

T

T

Rule #2 for Variances continued

We could similarly construct and

find that:

This illustrates Rule #2 on page 400.

***WITH INDEPENDENCE, variances

when sets

ADD

do

and

***STANDARD DEVIATIONS o n t

var var var var

X Y

X Y X Y X Y

X Y X Y

T

T T T T

T T

add.

American Roulette18 Black Numbers 18 Red Numbers 2 Green Numbers

Betting $1.00 on one number has a probability of 1/38 of winning $35.00. The probability you will lose your dollar is 37/38.

Your expectation is ($35)(1/38) – ($1)(37/38) = -$0.0526

The casino takes in $0.0526 for every $1 that is wagered on the game!!