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
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Discrete vs. Continuous
Random variables
Random Variables
Discrete
Examples:
Continuous
Examples:
Number of students
showed up next time
The temperature
tomorrow
Number of late apt.
rental payments in Oct.
The total rental payment
collected by Sep 30
Your score in this
coming mid-term exam
The expected lifetime
of a new light bulb
BUS304 – Probability Theory
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Discrete Probability Distribution
Table
Discrete
Probability
Distribution
X
P(X)
0
0.25
1
0.5
2
0.25
Graph
Probability
All the possible values of x
.50
.25
0
1
BUS304 – Probability Theory
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x
3
Exercise
Describe 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.
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Measures of Discrete
Random Variables
Expected 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
-2
0
2
P(x)
.25
.50
.25
E(x) = (-2) * 0.25 + 0 * 0.5 + 2 * 0.25
=0
Your expected gain is 0! – a fair game.
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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
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Exercise
You are working part time in a restaurant. The amount of tip you
get each time varies. Your estimation of the probability is as follows:
$ per night
Probability
50
0.2
60
0.3
70
0.4
80
0.1
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?
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More exercise
What 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)?
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Rule for expected value
If 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)
y
P(y)
x+y
P(x+y)
-1
0.5
-1
0.5
-2
0.25
1
0.5
1
0.5
0
0.5
2
0.25
E(x)=0
E(y)=0
BUS304 – Probability Theory
E(x+y)=0
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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))2P(x)
Step 6: sum up all the weighted square distance. =39.6+4.8+14.4+25.6=84.4
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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
 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 
variance using a relative
e.g. for the example on page 10
frequency table.
 Written as var(x) or  2
  84.4  9.19
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Homework
Problem 4.40
Problem 4.50
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