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Probability Distributions
Finite Random Variables
Probability Distributions
 Recall a Random Variable was:
 numerical value associated with the outcome of an
experiment that was subject to chance
 Examples:
 The number of heads that appear when flipping three coins
 The sum obtained when two fair dice are rolled
 Value of an attempted loan workout
Finite Random Variables
 Types of random variables:
 Finite Random Variable:
 Can only assume a countable number of values (i.e.
they can all be listed)
 Examples
 Flipping a coin three times & recording # of heads
 Rolling a pair of dice & recording the sum on each roll
Finite Random Variables
 Types of Random Variables (cont):
 Continuous Random Variable:
 Can assume a whole range of values which cannot be
listed:
 Examples:
 Arrival times of customers using an ATM at the start of the
hour
 Time between arrivals
 Length of service at an ATM machine
 We’ll first look at finite random variables
Finite Random Variables
 Ex: Suppose we flip a coin 3 times and record
each flip. Let X be a random variable that
records the # of heads we get.
S  HHH , HHT , HTH, HTT ,THH,THT ,TTH,TTT 
 What are the possible values of X?
Finite Random Variables
 X can be 0, 1, 2, or 3
 Notice that we can list all possible values of the
random variable X
 This makes X a finite random variable
Finite Random Variables
 Recall
P X  x 
 Told us the probability the random variable X
(upper case) assumed a certain value x (lower case)
 List all possible values of x (lower case) and
their probabilities in table
x
0
1
2
3
P (X=x )
0.125
0.375
0.375
0.125
Finite Random Variables
 Notice each value of the random variable has
one probability associated with it
 For a finite random variable we call this the
probability mass function
 Abbreviated (p.m.f.)
x
0
1
2
3
P (X=x )
0.125
0.375
0.375
0.125
Finite Random Variables
 Why a function?
 Each value of the random variable has exactly one
probability assigned to it
 Since this is a function, the following are equivalent
for a finite random variable:
fX x   P X  x 
Finite Random Variables
 Like any function, the p.m.f. for a finite random
variable has several properties:
 Domain: discrete numbered values
(e.g. {0, 1, 2,3} )
 Range:
 Sum:
0  fX x   1
 f x   1
X
All x
Finite Random Variables
 If we graph a p.m.f., should be a histogram with sum
of all heights equal to 1
P(X=x)
0.4
0.3
0.2
P(X=x)
0.1
0
0
1
2
3
X=x
 Each bar height corresponds to P(X=x)
Finite Random Variables
 Cumulative Distribution Function
 Abbreviated c.d.f.
 Determines probability of all events occurring up to
and including a specific event
 FX x   P X  x 
Finite Random Variables
 Ex: Find FX 0  from our coin problem. Do the
same for FX 2 and FX 1.7 
 Sol:
FX 0   P X  0   0.125
FX 1.7  P X  1.7  P X  0   P( X  1)  0.5
FX 2  P X  2  P X  0   P X  1  P X  2  0.875
Finite Random Variables
 Notice:
Interval
(-∞,0)
FX(x) = P(X≤x)
0
[0,1)
0.125
[1,2)
0.125 + 0.375=0.500
[2,3)
0.125+0.375+0.375=0.875
[3,∞)
0.125+0.375+0.375+0.125=1
Finite Random Variables
 The last table is describing a piece-wise
function:
 0
0.125

FX x   0.500
0.875

 1
if
x0
if 0  x  1
if 1  x  2
if
2 x3
if
x3
Finite Random Variables
 The graph of the c.d.f.
F(X)
-6
-5
-4
-3
-2
1.125
1
0.875
0.75
0.625
0.5
0.375
0.25
0.125
0
-1
0
X=x
1
2
3
4
5
6
Finite Random Variables
 Notice for the c.d.f. of a finite random variable
 Graph is a step-wise function
 Domain is all real #s
 Graph never decreases
 Approaches 1 as x gets larger
Finite Random Variables
 Ex. Use the sample space for rolling two dice to
graph P  X  x  .
 (1,1)
 ( 2 ,1)

 ( 3,1)
S
 ( 4 ,1)
 (5,1)

 ( 6,1)
(1, 2 )
(2, 2)
( 3, 2 )
(4, 2)
(5, 2 )
( 6, 2 )
(1, 3)
( 2 , 3)
( 3, 3)
( 4 , 3)
(5, 3)
( 6, 3)
(1, 4 )
(2, 4)
( 3, 4 )
(4, 4)
(5, 4 )
( 6, 4 )
(1, 5)
( 2 , 5)
( 3, 5)
( 4 , 5)
(5, 5)
( 6, 5)
(1, 6) ü
( 2 , 6) 

( 3, 6) 
ý
( 4 , 6) 
(5, 6) 

( 6, 6) þ
Finite Random Variables
 Soln.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
P (X =x )
0.0000
0.0278
0.0556
0.0833
0.1111
0.1389
0.1667
0.1389
0.1111
0.0833
0.0556
0.0278
0.0000
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
Finite Random Variables
 Ex. Find fX 3 in the previous example. Find fX 7 
in the previous example. Find fX 2.7 in the
previous example.
 Soln.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
P (X =x )
0.0000
0.0278
0.0556
0.0833
0.1111
0.1389
0.1667
0.1389
0.1111
0.0833
0.0556
0.0278
0.0000
fX 3  0.0556
fX 7   0.1667
fX 2.7   0
Finite Random Variables
 Remember: Cumulative distribution function
adds probabilities up to and including a certain
value
 Ex. FindFX 3  in the previous example. Find FX 7 
in the previous example. Find FX 2.7  in the
previous example.
Finite Random Variables
 Soln.
x
1
2
3
4
5
6
7
8
9
10
11
12
13
P (X =x )
0.0000
0.0278
0.0556
0.0833
0.1111
0.1389
0.1667
0.1389
0.1111
0.0833
0.0556
0.0278
0.0000
FX 3  P X  3
 0  0.0278  0.0556
 0.0833
FX 7  P X  7
 0  0.0278  ...  0.1667
 0.5833
FX 2.7  P X  2.7
 0  0.0278
 0.0278
Finite Random Variables
• Sample c.d.f.
Cummulative Distribution Function
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
1
2
3
4
5
6
7
8
9
10
Finite Random Variables
 Notice the graph starts at height 0
 Notice the graph ends at height 1
 The graph “steps” to next height
 The size of each “step” corresponds to the value
of the p.m.f. at that x-value
Finite Random Variables
 The bullet on the last slide gives us the ability
to:
p.m.f.
c.d.f
Finite Random Variables
 Binomial Random Variable
 Special type of finite r.v.
 Collection of Bernoulli Trials
 Bernoulli Trial is an experiment with only 2 possible
outcomes
 Each trial is independent
Finite Random Variables
 Excel has a built in Binomial R.V. function
 BINOMDIST
Finite Random Variables
 Ex: Suppose we flip a biased coin whose probability of
landing heads is 0.7. Let’s say we do this 3 times and
record each flip. Let X be a random variable that
records the # of heads we get.
S  HHH , HHT , HTH, HTT ,THH,THT ,TTH,TTT 
 This is an example of a binomial random variable
Finite Random Variables
 In Excel:
X=x
0
1
2
3
P(X=x)
0.027
0.189
0.441
0.343
 This is the p.m.f
How many
times you
perform the
experiment
Value of Random
Variable
Probability of
success on
each trial
FALSE (p.m.f) ;
TRUE (c.d.f)
Finite Random Variables
 In Excel:
X=x
0
1
2
3
 This is the c.d.f
F(X)
0.027
0.216
0.657
1
Finite Random Variables
 Ex. Historically, 83% of all students pass a
particular class. If there are 34 students in the
class, what is the probability that exactly 28 will
pass? What is the probability that at least 28
students pass? What is the probability that at
most 28 students pass?
Finite Random Variables
 What is the probability that exactly 28 students pass?
fX 28   P X  28 
Soln: 0.1760
Finite Random Variables
 What is the probability that at least 28 students pass?
P X  28   1  P X  28 
 1  P X  27 
 1  FX 27 
Soln: 1 - 0.3545
0.6455
Finite Random Variables
 What is the probability that at most 28 students pass?
P  X  28   FX x 
Soln: 0.5305
Finite Random Variables
 Sample p.m.f.
Probability Mass Function
0.2
0.16
0.12
0.08
0.04
0
0
4
8
12
16
20
24
28
32
Finite Random Variables
 Mean of a finite random variable
 same as expected value in project 1
 add product of each value and it’s respective
probability
X  E X  
 x  f x 
X
all possible x
Finite Random Variables
 Ex. Historically, 83% of all students pass a
particular class. If there are 34 students in the
class, find the mean number of students that
will pass.
Soln. Approximately 28.22 students will pass
(Two ways this can be found)
X=x
Finite Random Variables
 Method 1:
 Use BINOMDIST to make
p.m.f.
 Then use:
X  E X  
 x  f x 
X
all possible x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
P(X=x)
6.84326E-27
1.13598E-24
9.15134E-23
4.76587E-21
1.80332E-19
5.28267E-18
1.24661E-16
2.43455E-15
4.01164E-14
5.65825E-13
6.90639E-12
7.35696E-11
6.88453E-10
5.68831E-09
4.16585E-08
2.71188E-07
1.5723E-06
8.12806E-06
3.74794E-05
0.000154095
0.000564259
0.001836607
0.005298661
0.013497356
0.030203643
0.058985939
0.099688905
0.144212272
0.176023803
0.177809034
0.144687744
0.091150533
0.041721476
0.012345392
0.001772781
E(X)
x*P(X=x)
0
1.14E-24
1.83E-22
1.43E-20
7.21E-19
2.64E-17
7.48E-16
1.7E-14
3.21E-13
5.09E-12
6.91E-11
8.09E-10
8.26E-09
7.39E-08
5.83E-07
4.07E-06
2.52E-05
0.000138
0.000675
0.002928
0.011285
0.038569
0.116571
0.310439
0.724887
1.474648
2.591912
3.893731
4.928666
5.156462
4.340632
2.825667
1.335087
0.407398
0.060275
28.22
Finite Random Variables
 Method 2:
 There is a special formula that ONLY works for
random variables that are binomially
distributed:
X  E X   n  p