BUS304 – Probability Theory 1

Probability Distribution Random Variable:

A variable with random (unknown) value.

Examples

1. Roll a die twice: Let x be the number of times 4 comes up.

x = 0, 1, or 2

2. Toss a coin 5 times: Let x be the number of heads x = 0, 1, 2, 3, 4, or 5

3. Same as experiment 2: Let’s say you pay your friend $1 every time head shows up, and he pays you $1 otherwise. Let x be amount of money you gain from the game.

What are the possible values of x?

BUS304 – Probability Theory 2

Discrete vs. Continuous Random variables

Random Variables

ContinuousDiscrete

Examples:

Number of students showed up next time

Number of late apt. rental payments in

Oct.Your score in this

coming mid-term exam

Examples:

The temperature tomorrow

The total rental payment collected by Sep 30

The expected lifetime of a new light bulb

BUS304 – Probability Theory 3

Discrete Probability Distribution

Discrete Probability Distribution

Table

Graph

X P(X)

0 0.25

1 0.5

2 0.25

0 1 2 x

.50

.25 P

rob

abil

ity

All the possible values of x

BUS304 – Probability Theory 4

ExerciseDescribe the probability distribution of the

following experiments:

Draw a pair of dice, x is the random variable representing the sum of the total points.

In a community with 100 households, 10 do not have kid, 40 have just one kid, 30 have 2 kids, and 20 have 3 kids. Randomly select one household. Let X be the number of kids in the household.

BUS304 – Probability Theory 5

Measures of Discrete Random VariablesExpected value of a discrete distribution

An weighted average, taking into account the probability

The expected value of random variable x is denoted as E(x)

E(x)= xi P(xi)E(x)= x1P(x1) +x2P(x2) + … + xnP(xn)

Example: What is your expected gain when you play the flip-coin game twice?

x P(x)

-2 .25

0 .50

2 .25

E(x) = (-2) * 0.25 + 0 * 0.5 + 2 * 0.25

= 0Your expected gain is 0! – a fair game.Your expected gain is 0! – a fair game.

BUS304 – Probability Theory 6

Worksheet to compute the expected value Step1: develop the distribution table according to the description of

the problem.

Step2: add one (3rd) column to compute the product of the value

and the probability

Step3: compute the sum of the 3rd column The Expected Value

x P(x) x*P(x)

-2 0.25 -2*.25=-0.5

0 0.5 0*0.5=0

2 0.25 2*0.25=0.5

E(x) =-0.5+0+0.5=0

BUS304 – Probability Theory 7

ExerciseYou are working part time in a restaurant. The amount of tip you get each time varies. Your estimation of the probability is as follows:

You bargain with the boss saying you want a more fixed income. He said he can give you $62 per night, instead of letting you keep the tips. Would you want to accept this offer?

$ per night Probability

50 0.2

60 0.3

70 0.4

80 0.1

BUS304 – Probability Theory 8

More exerciseWhat is the expected gain if you plan the flip

coin game just once? Three times? Four Times?

What is the expected number of kid in that community (see the example on page 4)?

BUS304 – Probability Theory 9

Rule for expected valueIf there are two random variables, x and y. Then

E(x+y) = E(x) + E(y) Example:

• x is your gain from the flip-coin game the first time• y is your gain from the flip-coin game the second time• x+y is your total gain from playing the game twice.

x P(x)

-1 0.5

1 0.5

y P(y)

-1 0.5

1 0.5E(x)=0

E(y)=0

x+yP(x+y

)

-2 0.25

0 0.5

2 0.25

E(x+y)=0

BUS304 – Probability Theory 10

Measures – variance Variance: a weighted average of the squared deviation from the

expected value.

x P(x) x – E(x) (x-E(x))2 (x-E(x))2P(x)

50 0.2 50-64=-$14 (-14)2=196 196*0.2=39.6

60 0.3 -$4 16 4.8

70 0.4 $6 36 14.4

80 0.1 $16 256 25.6

Step 1: develop the probability distribution table.

Step 2: compute the mean E(x): 50x0.2+60x0.3+70x0.4+80x0.1=64

Step 3: compute the distance from the mean for each value (x-E(x))

Step 4: square each distance (x – E(x))2

Step 5: weight the squared distance: (x-E(x))2 P(x)

Step 6: sum up all the weighted square distance. =39.6+4.8+14.4+25.6=84.4

BUS304 – Probability Theory 11

Variance and Standard deviation

variance

The variance of a random

variable has the same

meaning as the variance of

population

Calculation is the same as

calculating population

variance using a relative

frequency table.

Written as var(x) or

Standard deviation of a random variable: Same of the population

standard deviation

Calculate the variance

Then take the square root of the variance.

Written as SD(x) or

e.g. for the example on page 10

2

84.4 9.19

BUS304 – Probability Theory 12

HomeworkProblem 4.40Problem 4.50