Download Probability Distributions: Finite Discrete Random Variables

Probability
Distributions: Finite
Random Variables
Random Variables

A random variable is a numerical value
associated with the outcome of an experiment.
 Examples:



The number of heads that appear when flipping three coins
The sum obtained when two fair dice are rolled
In both examples, we are not as interested in the
outcomes of the experiment as we are in
knowing a number value for the experiment
Example—Flipping a Coin 3 Times
Suppose that we flip a coin 3 times and
record each flip.
 S = {HHH, HHT, HTH, HTT, THH, THT,
TTT, TTT}
 Let X be a random variable that records
the # of heads we get for flipping a coin 3
times.

Example (cont)

The possible values our random variable X can
assume:
X

= x where x = {0, 1, 2, 3}
Notice that the values of X are:
 Countable, i.e. we can list all possible
 The values are whole numbers


values
When a random variable’s values are countable
the random variable is called a finite random
variable.
And when the values are whole #’s the random
variable is discrete.
Probabilities
Just as with events, we can also talk about
probabilities of random variables.
 The probability a random variable
assumes a certain value is written as

 P(X

=x)
Notice that X is the random variable for the
# of heads and x is the value the variable
assumes.
Probabilities



We can list all the probabilities for our
random variable in a table.
The pattern of probabilities for a
random variable is called its
probability distribution.
For a finite discrete random
variable, this pattern of probabilities is
called the probability mass function
( p.m.f ).
X=x
P(X=x)
0
1/8
1
3/8
2
3/8
3
1/8
Probability Mass Function


We consider this table to
be a function because
each value of the random
variable has exactly one
probability associated
with it.
Because of this we use
function notation to say:
f X x   P X  x 
X=x
0
1
2
3
P(X=x)
1/8
3/8
3/8
1/8
Properties of Probability Mass
Function

Because the p.m.f is a function it has a
domain and range like any other function
you’ve seen:
 Domain:
{all whole # values random variable}
 Range:
0  f X x   1
 Sum:
 f x   1
X
All x
Representing the p.m.f.


Because the p.m.f function uses only whole #
values in its domain, we often use histograms to
show pictorially the distribution of probabilities.
Here is a histogram for our coin example:
P(X=x)
0.4
0.3
0.2
P(X=x)
0.1
0
0
1
2
X=x
3
Things to notice:
The height of each rectangle corresponds
to P(X=x)
 The sum of all heights is equal to 1

P(X=x)
0.4
0.3
0.2
P(X=x)
0.1
0
0
1
2
X=x
3
Cumulative Distribution Function


The same probability information is often given
in a different form, called the cumulative
distribution function, c.d.f.
Like the p.m.f. the c.d.f. is a function which we
denote as Fx(x) (upper case F) with the following
properties:
 Domain:
the set of all real #s
 Range: 0≤ Fx(x) ≤1
 Fx(x) = P(X≤x)
 As x →∞, Fx(x) →1 AND As x →-∞, Fx(x) →0
Graphing the c.d.f.
Let’s graph the c.d.f. for our coin example.
 According to our definitions from the
previous slide:

 Domain:
the set of all real #s
 Range: 0≤ Fx(x) ≤1
 Fx(x) = P(X≤x)
Graphing (cont)



Here’s the p.m.f. :
X=x
0
1
2
3
P(X=x) 1/8
3/8
1/8
3/8
Because the domain for the cdf is the set of all
real numbers, any value of x that is less than
zero would mean that Fx(x) is 0 since there is no
way for a flip of three coins to have less than 0
heads. The probability is zero!
Also, because the number of heads we can get
is always at most 3, Fx(x) = 1 when x ≥ 3.
Graphing (cont)




Now we need to look at what happens for the
other values.
If 0≤ x <1, then Fx(x) = P(X ≤ x) = P(X=0) = 1/8
If 1≤ x <2, then Fx(x) = P(X ≤ x) = P(X=1)
+P(X=0)= 1/8+3/8=4/8
If 2≤ x <3, then Fx(x) = P(X ≤ x) =
P(X=2)+P(X=1) +P(X=0)= 1/8+3/8+3/8=7/8
The c.d.f.

All of the previous information is best
summarized with a piece-wise function:
0
1

 84
FX  x   
8
7
8
 1
if
x0
if
0  x 1
if
1 x  2
if
2 x3
if
x3
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
Things to notice






The graph is a step-wise function. This is
typically what you will see for finite discrete
random variables.
Domain: the set of all real #s
Range: 0≤ Fx(x) ≤1
Fx(x) = P(X≤x)
As x →∞, Fx(x) →1 AND As x →-∞, Fx(x) →0
At each x-value where there is a jump, the size
of the jump tells us the P(X=x). Because of this,
we can write a p.m.f. function from a c.d.f.
function and vice-versa
Expected Value of Finite Discrete
Random Variable

Expected Value of a Discrete Random Variable
is
 x  P( X  x)
all x

Note, this is the sum of each of the heights of
each rectangle in the p.m.f., multiplied by the
respective value of X in the pmf.
Example

Box contains four $1 chips, three $5 chips,
two $25 chips, and one $100 chip. Let X
be the denomination of a chip selected at
random. The p.m.f. of X is displayed
below.
X
$1.00
$5.00
$25.00
$100.00
fX(X)
0.40
0.30
0.20
0.10
Questions
What is P(X=25)?
 What is P(X≤25)?
 What P(X≥5)?
 Graph the c.d.f.
 What is the E(X)?

Answers
P( X  $25)  f X ($25)  0.2
P ( X  $25)  P ( X  $1)  P ( X  $5)  P ( X  $25)
 f X ($1)  f X ($5)  f X ($25)
 0.4  0.3  0.2
 0.9
P( X  $25)  FX ($25)  0.9
P( X  $5)  f X ($5)  f X ($25)  f X ($100)
P X  5
 0.3  0.2  0.1
 0.6
 1  P X  5  1  P X  1
 1  FX 1  1  0.4  0.6
Answer(CDF)
Cumulative Distribution Function
1.2
1.0
FX (x )
0.8
0.6
0.4
0.2
0.0
-20
0
20
40
60
x
80
100
120
Expected Value
x
x  f X (x)
f X (x)
$
1
0.4
0.40
$
5
0.3
1.50
$ 25
0.2
5.00
$ 100
0.1
10.00
Sum
1.0
X 
$16.90
Bernoulli Random Variables
Bernoulli Random Variables are a special
case of discrete random variables
 In a Bernoulli Trial there are only two
outcomes: success or failure

Bernoulli Random Variable



Let X stand for the number of successes in n Bernoulli
Trials, X is called a Binomial Random Variable
Binomial Setting:
1. You have n repeated trials of an experiment.
2. On a single trial, there are two possible outcomes,
success or failure.
3. The probability of success is the same from trial to
trial.
4. The outcome of each trial is independent.
Expected Value of a Binomial R.V. is
E(X)=np, p is probability of success
Loaded Coin
Suppose you have a coin that is biased
towards heads. Let’s suppose that on any
given flip of the coin your get heads about
60% of the time and tails 40% of the time.
 Let X be the random variable for the
number of heads obtained in flipping the
coin three times.

Loaded Coin



S = {HHH, HHT, HTH, HTT, THH, THT, TTT,
TTT}
A “success” in this experiment will be the
occurrence of a head and a “failure” will be when
we get a tail.
Because getting a head is more likely now, we
need to look at what the probability is for getting
each of the outcomes in our sample space.
Loaded coin


For example: What is the probability of getting
HHH?
Because each trial or flip is independent, we can
say that:
PH  H  H   PH PH PH   0.600.600.60  0.216

Similarly, we can also ask what the probabilities
are for other outcomes in our experiment.
Loaded Coin
Outcome
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Probability
(0.60)(0.60)(0.60)=0.216
(0.60)(0.60)(0.40)=0.144
(0.60)(0.40)(0.60)=0.144
(0.60)(0.40)(0.40)=0.096
(0.40)(0.60)(0.60)=0.144
(0.40)(0.60)(0.40)=0.096
(0.40)(0.40)(0.60)=0.096
(0.40)(0.40)(0.40)=0.064
Loaded Coin: p.m.f.
X=x
P(X=x)
0
0.064
1
0.288
2
0.432
3
0.216
Loaded Coin: Graph of pmf
PMF
P(X=x)
0.5
0.4
0.3
Series1
0.2
0.1
0
0
1
2
X=x
3
Your Turn!
Graph the cdf of our biased coin example
 Excel: BINOMDIST

F(x)
1.2
1
0.8
0.6
0.4
0.2
0
-5
-3
-1
1
X
3
5