Download Notes on Probability Distributions

Random Variables
OBJECTIVE
Construct a probability
distribution.
 Find measures of
center and spread for a
probability distribution.

RELEVANCE
To find the likelihood of
all possible outcomes of
a probability distribution
and to describe the
distribution.
Definition……

Random Variable – a random
variable, x, represents a numerical
value associated with each outcome of
a probability experiment; values are
determined by chance (Mutually
exclusive)
Example……

You have 4 True or
False questions and
you observe the
number correct.


Random Variable (x)
# correct
Possible Values of x
0, 1, 2, 3, 4
Example……

Count the number of
siblings in your
family.


Random Variable (x)
# of siblings
Possible Values
0, 1, 2, 3, …..
Example……

Toss 5 coins and
observe the number
of heads.


Random Variable (x)
# of heads
Possible Values
0, 1, 2, 3, 4, 5
2 Types of Random
Variables……


Discrete – can be counted
Ex: # of joggers
Continuous – can be measured
Ex: Height, Weight, Temp, Time,
Distance
Probability Distributions –
Discrete ……
Probability Distribution……


Consists of the values a random
variable can assume and the
corresponding probabilities of the
variables.
The probabilities used are theoretical.
Example……

Construct a
probability
distribution of
tossing 2 coins and
getting heads.

Remember:
x = the number of
heads possible
p(x) = the
probability of getting
those heads
Answer……

The sample space
for tossing 2 coins
consists of
TT
HT
TH
HH

The probability
distribution of
getting heads…
x
0
1
2
P(x)
1/4
2/4
1/4
Now Graph It……
2
4
P (x )
1
4
0
1
x
2
Example……

Construct a
probability
distribution of
tossing 3 coins and
getting heads.

Remember:
x = the # of heads
possible
P(x) = the
probability of getting
those heads
Answer……

The sample space of
tossing 3 coins
consists of
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT

The probability
distribution….
x
0
1
2
3
P(x)
1/8
3/8
3/8
1/8
Graph It……
3
8
P (x )
2
8
1
8
0
1
2
x
3
Example……

Construct a
probability
distribution for
rolling a die.

The probability
distribution….
x
1
2
3
4
5
6
P(x)
1/6
1/6
1/6
1/6
1/6
1/6
2 Requirements for a
Probability Distribution……


1. 0  P( x)  1
Probability has to
fall between 0 and 1
for individual
probabilities


2.  P( x)  1
All the P(x)’s must
add up to 1.
Is it a probability distribution?
Example 1……
x
0
5
10 15 20
P(x) 1/5 1/5 1/5 1/5 1/5
YES
Example 2……
x
1
2
3
4
P(x)
1/4
1/8
1/16
9/16
4/16
2/16
YES
Example 3…..
x
2
3
7
NO
 P( x)  1
P(x)
0.5
0.3
0.4
Example 4…..
x
0
2
P(x) -1.0 1.5
2
6
0.3
0.2
NO
Individual P(x) does
not fall between 0
and 1
Complete the Chart……
1 – 0.72 = 0.28
x
1
2
3
4
5
P(x)
0.2
0.35
0.1
0.07
?
Probability Functions
Probability Function……

A rule that assigns probabilities to the
values of the random #’s.
Example……

The following function,
is a probability distribution for
x = 1, 2, 3, 4. Write the probability
distribution.
x
P( x) 
10
Answer……

x
1
2
3
4
P(x)
1/10
2/10
3/10
4/10
You can set a
formula in the lists
using your calculator
by putting x’s in L1
and setting L2 as
L1/10……or you can
just plug it in by
hand.

Would the function
x
P( x) 
10

Be a probability distribution for
x= 2, 3, 4, 5?
Answer……


Here is the
distribution……
The sum of the P(x)
values adds up to a
number greater than
1….therefore it is NOT
a probability
distribution.
x
2
3
4
5
P(x)
2/10
3/10
4/10
5/10
Example……

Write the probability
distribution for the
function y  x
3
if x = 0, 1, 2
x
0
1
2
P(x)
0
1/3
2/3
Mean, Standard Deviation, and
Variance of a Probability
Distribution
Section 5.3

Take Note: Probability distributions
may be used to represent theoretical
populations, therefore, we will use
population parameters and our symbols
used will be for population values.
  mean
 2  var iance
  s tan dard deviation
Mean :   [ x  P( x)]
Variance :   [( x   )  P( x)]
2
St. Deviation :   
2
2
Example……



A. Find the mean,
variance, and
standard deviation.
B. Find the
probability that
x is between   2
C. Find the
probability that
x is between   
x
P(x)
0
0.2
1
0.1
2
0.3
3
0.4
Mean:
  [ x  P( x)]

x
P(x)
0
0.2
1
0.1
2
0.3



3
0.4
We are going to do
this on our graphing
calculator.
Put x’s in L1 and
P(x) in L2.
Set a formula in L3:
L1xL2
The sum of this
column is your
mean.
This is what it looks like on the
calculator.
The mean is 1.9
Another way to sum a list…..



2nd Stat
Math
Sum L(#)
This is actually better because you
can store the mean to a location
Variance: 


2
 [x     P( x)]
You’ll need to add a
L4 in your
calculator.
Set a formula to find
the formula above.
2

The sum of L4 is
your variance.
The var iance is 1.29.
Standard Deviation:   

2
The variance was 1.29.
 
2
  1.29
  1.14
The s tan dard deviation is 1.14.
Find the probability that
x is between   2

Remember that


Remember that

  1.9.
  1.14
Find the x’s for which
you should sum their
probabilities:
The x’s will be between
1.9 – 2(1.14) = -0.38
1.9 + 2(1.14) = 4.18
Remember: The x values are
between -0.38 and 4.18.

x
P(x)
x[P(x)]
Variance
Formula
0
0.2
0
0.722

1
0.1
0.1
0.081
2
0.3
0.6
0.003
3
0.4
1.2
0.484


All of our x’s fall in
between these 2
values.
Add the probabilities
that goes along with
these values.
0.2+0.1+0.3+0.4 =
1
The answer is 1.
Find the probability that
x is between   

Remember that

Remember that
  1.9.
  1.14


Find the x’s for
which you should
sum their
probabilities:
The x’s will be
between
1.9 – 1.14 = 0.76
1.9 + 1.14 = 3.04
Remember: The x values are
between 0.76 and 3.04.

x
P(x)
x[P(x)]
0
0.2
0
Variance
Formula
0.722

1 0.1
0.1
0.081

2 0.3
0.6
0.003
3 0.4
1.2
0.484

Only 3 of the x
values fall between
the values listed
above.
Add the probabilities
that go with those 3.
0.1+0.3+0.4 = 0.8
The answer is 0.8.
Example……

A. Set up the distribution for
x
P( x) 
for x  1, 2, 3
6


B. Find the mean, variance, and standard
deviation of the probability distribution.
C. Find the probability that x is between   
Set up the distribution……
x
P(x)
1
1/6
2
2/6
3
3/6
Mean:
  [ x  P( x)]

x
P(x)

1
1/6

2
2/6

3
3/6
We are going to do
this on our graphing
calculator.
Put x’s in L1 and
P(x) in L2.
Set a formula in L3:
L1xL2
The sum of this
column is your
mean.
This is what it looks like on the
calculator……
The mean is 2.3
Variance:



 2  [x   2  P( x)]
You’ll need to add a
L4 in your
calculator.
Set a formula to find
the formula above.
The sum of list 4 is
the variance.
The VARIANCE is 0.56.
Standard Deviation:   

The variance was 0.555555555.
  2
  0.555555555
  0.75

The standard deviation is 0.75.
2
Find the probability that
x is between   

Remember that

  2.33

Remember that
  0.75

Find the x’s for
which you should
sum their
probabilities:
The x’s will be
between
2.33 – 0.75 = 1.58
2.33 + 0.75 =3.08
Remember: The x values are
between 1.58 and 3.08.

x
P(x)
x[P(x)]
Variance
Formula

1
1/6
0.16667
0.2963

2 2/6
0.66667
0.03704
3 3/6
1.5
0.22222

Only 2 of the x
values fall between
1.58 and 3.08.
Add the probabilities
of those 2 x-values.
2/6 + 3/6 = 5/6
The answer is 5/6.

Worksheet