2.1
Discrete probability distributions
2.
Discrete Probability Distributions
G
Counting methods
G.1
Permutations and combinations
Task:
1.
Evaluate
10!
2.
20P
8
3.
18C
11
Factorials (!), permutations(n Pr) and combinations (nCr) are
located in the MATH(ematics) menu under PRB (an abbreviation
for Probability).
Step 1. Evaluate 10!
Start on the HOME screen (press 2nd [QUIT] if you are not on
the HOME screen), and CLEAR .
(1) Type in 10.
(2) Press MATH
and use the arrow key to move across to the
PRB menu and the down arrow to move to option 4:!.
(3) Press ENTER to select. This pastes the factorial function
onto the HOME screen.
(4) Finally, press ENTER
to evaluate.
Thus,
10! = 3628800
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.2
Discrete probability distributions
Step 2. Evaluate 20P8
Start on the HOME screen (press 2nd [QUIT] if you are not on
the HOME screen), and CLEAR .
(1) Type in 20.
(2) Press MATH
and use the arrow key to move across to the
PRB menu and the down arrow to move to option 2: nP r .
(3) Press ENTER to select. This pastes the permutation
function onto the HOME screen.
(4) Finally, type in 8 and press ENTER
to evaluate.
Thus,
20P = 5079110400
8
Step 3. Evaluate 18C11
Start on the HOME screen (press 2nd [QUIT] if you are not on
the HOME screen), and CLEAR .
(1) Type in 18.
(2) Press MATH
and use the arrow key to move across to the
PRB menu and the down arrow to move to option 3: nCr.
(3) Press ENTER to select. This pastes the permutation
function onto the HOME screen.
(4) Finally, type in 11 and press ENTER
to evaluate.
Thus,
18C
11 = 31824
Exercises
1.
Evaluate each of the following by hand and check your answer with the calculator:
(1)
2.
6!
(2)
6P
4
(3)
8C
5
Use your calculator to show that:
(1)
13! = 6227020800
(3)
35C
(2)
15P = 1816214400
9
22 = 1476337800
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.3
H
The binomial probability distribution
H.1
Determining binomial probabilities
Discrete probability distributions
A discrete random variable X is said to have a binomial distribution if
Pr(X=x) = nC x px(1–p)n-x
for x= 0,1, 2,...
The binomial distribution is defined by two parameters, n , the number of independent
trials and p, the probability of success for any one of the trials.
Note:
To indicate that a random variable has a binomial distribution with parameters n and p
we write X ~Bi(n, p ).
Task: If X~Bi(20, 0.6), evaluate
1.
Pr(X=8) 2.
Pr(X≤11)
3.
Pr(X≥5)
4.
Pr(2≤X≤14)
Step 1. Evaluate Pr(X=8) given X~Bi(20,0.6)
Start on the HOME screen (press 2nd [QUIT] if you are not on
the HOME screen), and CLEAR .
(1) Go to the [DISTR] menu ( 2nd [DISTR] ( VARS ) ) and use
the arrow key to move down the DISTR menu to option
0: binompdf(. Select by pressing ENTER . This pastes the
binompdf( (binomial probability distribution function) command
onto the HOME screen.
(2) Complete the command as follows:
binompdf(20,0.6,8)
the number of trials, n
number of successes, x
probability of success, p
(3) Finally, press ENTER
to evaluate.
Thus,
Pr(X=8)= 0.0355 (correct to 4 decimal places).
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.4
Discrete probability distributions
Step 2. Evaluate Pr(X ≤11) given X~Bi(20,0.6)
Start on the HOME screen (press 2nd [QUIT] if you are not on
the HOME screen), and CLEAR .
(1) Go to the [DISTR] menu ( 2nd [DIST] ( VARS )) and use
the arrow key to move down the DISTR menu to option
A:binomcdf(. Select by pressing ENTER . This pastes the
binomcdf( (binomial cumulative probability distribution function)
command onto the HOME screen.
(2) Complete the command as follows:
binomcdf(20,0.6,11)
the number of trials, n
probability of success, p
(3) Finally, press ENTER
upper limit for the
number of successes, x
to evaluate.
Thus,
Pr(X≤11)= 0.4044 (correct to 4 decimal places).
Step 3. Evaluate Pr(X ≥5) given X~Bi(20,0.6)
The calculator cannot evaluate this probability directly, but we can
make use of the fact that
Pr(X≥5) = 1 – Pr(X≤4)
and proceed as follows.
Start on the HOME screen (press 2nd [QUIT] if you are not on
the HOME screen), and CLEAR .
(1) Type in
1-
(2) Go to the DISTR menu ( 2nd [DISTR] ( VARS ) ) and use
the arrow key to move down the DISTR menu to option A:
binomcdf( Select by pressing ENTER . This pastes the
binomcdf( command onto the HOME screen.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.5
Discrete probability distributions
(3) Complete the command by adding as follows:
1-binomcdf(20,0.6,4)
the number of trials, n
upper limit for the
probability of success, p number of failures, x
(4) Finally, press ENTER
to evaluate.
Thus,
Pr(X≥5) = 0.9997 (correct to 4 decimal places).
Step 4.Evaluate Pr(2≤X≤14) given X~Bi(20,0.6)
The calculator cannot evaluate this probability directly, but we can
make use of the fact that
Pr(2≤X≤14) = Pr(X≤14)– Pr(X≤1)
and proceed as before.
Thus
Pr(2≤X≤14) = 0.8744 (correct to 4 decimal places).
Exercises
1.
If X~ Bi(8, 0.4), show that (correct to 4 decimal places):
(1)
(3)
(5)
2.
Pr(X=4) = 0.2322
Pr(X≥7) = 0.0085
Pr(X<6) = Pr( X≤5) =0.9502
(2)
(4)
Pr(X≤1) = 0.1064
Pr(1≤X≤3) = 0.5773
If X~ Bi(25, 0.2), use your calculator to show that, correct to 4 decimal places:
(1)
(3)
(5)
Pr(X=4) = 0.1867
Pr(X≥7) = 0.2200
Pr(X>4) = 1–Pr(X≤4) =0.5793
(2)
(4)
Pr(X≤10) = 0.9944
Pr(5≤X≤8) = 0.5326
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.6
H.2
Displaying binomial probability distributions
Task:
If X~ Bi(12, 0.6), display the distribution of X graphically.
Discrete probability distributions
To display a probability distribution graphically, we need to create two lists of values:
• the first containing the values taken by X, in this case 0, 1, 2, 3, ....12.
• the second containing the corresponding probabilities.
Step 1.
Go the stat list editor ( STAT [EDIT]) and create a list named X.
Enter the values 0,1, 2,...12 into this list.*
*If you are unsure how to do this, see section A.1.
or
If you are an experienced user, you can achieve the same result by
entering the following expression
seq(X,X ,0 ,12) STO˛
LX
on the HOME screen.
Step 2.
Next, create a list name PROB which will contain the probabilities.
You can now automatically enter the required binomial
probabilities as follows:
(1) Move the cursor to cover the word PROB at the top of the list
and press ENTER .
(2) Press 2nd [DISTR] ( VARS ) to take you to the [DISTR]
menu and use the arrow key to move down to option O:
binompdf( and press ENTER . This will paste the command
into the stat list editor opposite PROB=.
(3) Complete the command as follows:
PROB=binompdf(12,0.6)
the number of trials, n
probability of success, x
(4) Press ENTER and the required probabilities will now be
entered into the list called PROB.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.7
Discrete probability distributions
Step 3.
Go to the STATPLOT menu and set it up to display a scatterplot
with the probabilities on the vertical axis (y-axis) and x-values on
the horizontal axis (x-axis). Display by pressing ZOOM
9 *.
*If you are unsure how to do this, see section E.1.
If you have followed this procedure properly, you should have the
probability distribution as shown. Individual probabilities can be
read off the graph by pressing TRACE and using the horizontal
arrow keys to move from point to point.
Exercises
1.
If Y~ Bi(10, 0.4) display the distribution of Y graphically.
2.
Simultaneously plot the distributions of the random variables Y1, Y2, Y3 where:
Y1~ Bi(10, 0.1) use PLOT 1 with the points shown by a ’
Y2~ Bi(10, 0.5) use PLOT 2 with the points shown by a +
Y3~ Bi(10, 0.9) use PLOT 3 with the points shown by a .
Comment on how changing the value of p changes the shape of the distribution.
As p increases, the peak of the distribution moves to the right.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.8
Discrete probability distributions
I
The geometric probability distribution
I.1
Determining geometric probabilities
A discrete random variable X is said to have a geometric distribution if
Pr(X=x) = p(1–p) x-1
for x= 1, 2,..
The geometric distribution is defined by a single parameter, p.
Task:
If X~ Ge(0.5), evaluate
1.
Pr(X=4) 2.
Pr(X≤6) 3.
Pr(X≥3)
4.
Pr(2≤X≤5)
Geometric probabilities are evaluated through the DISTR(ibution)
menu. This is accessed by pressing 2nd [DISTR] ( VARS )
and selecting either:
option
D:geometpdf( to evaluate probabilities like Pr(X=x) or
option
E:geometcdf( to evaluate probabilities like Pr(X≤ x)
Using the calculator we see that, if X~ Ge(0.5):
1.
Pr(X=4) = geometpdf(0.5,4) = 0.0625
2.
Pr(X≤6) = geometcdf(0.5,6)= 0.984375
3.
Pr(X≥3)
= 1– Pr(x≤2)
= 1–geometcdf(0.5,2)
= 0.25
4.
Pr(2≤X≤5)
= Pr(X≤5)–Pr(X≤1)
= geometcdf(0.5,5)–geometcdf(0.5,1)
= 0.46875
Exercise
1.
If X~ Ge(0.2), use your calculator to show that, correct to 4 decimal places:
(1)
(3)
Pr(X=5) = 0.0819
Pr(X≥7) = 0.2621
(2)
(4)
Pr(X≤8) = 0.8322
Pr(3≤X≤9) = 0.5058
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.9
Discrete probability distributions
I.2
Displaying geometric probability distributions
Task:
If X~ Ge(0.2), display the distribution of X graphically for x= 1, 2, ...8.
To display a probability distribution graphically, we need to create two lists of values:
• the first containing the values taken by X, in this case 1, 2, 3, ....8.
• the second containing the corresponding probabilities.
Step 1.
Go the stat list editor and create a list named X. Enter the values
1, 2,...8 into this list.*
*If you already have a list called X, either:
1. simply clear it by moving the cursor onto the X at the top of the column
and press CLEAR and re-enter the required values,
or
2. edit it by deleting unwanted items or adding extra items.
Step 2.
Next, create a list name PROB which will contain the geometric
probabilities for x= 1, 2....8.
*If you already have a list called PROB, simply clear it by moving the
cursor onto the X at the top of the column and press CLEAR and
proceed as follows.
Return to the HOME screen ( 2nd [QUIT]) and enter the
following
geometpdf(0.2 ,L X ) ¿ L PROB
and press ENTER .*
* Get LX and LPROB from the list menu.
If you return to the stat list editor( STAT
ENTER ), you will see
that the required probabilities have been entered into the list
named PROB.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.10
Discrete probability distributions
Step 3.
Go to the STATPLOT menu and set it up to display a scatterplot
with the probabilities on the vertical axis (y-axis) and x-values on
the horizontal axis (x-axis). Making sure that all other statistical
plots are turned off, display the geometric probabilities by
pressing ZOOM
9 *.
*If you are unsure how to do this, see section E.1.
If you have followed this procedure properly, you should have the
probability distribution as shown. Individual probabilities can be
read off the graph by pressing TRACE and using the horizontal
arrow keys to move from point to point.
Exercises
1.
If X~ Ge(0.4) display the distribution of X graphically for x = 1, 2, .....8.
2.
Simultaneously plot the distributions of the random variables Y1, Y2, for x= 1, 2, ...10, where:
Y1~ Ge(0.1)
use PLOT 1 with the points shown by a
Y2~ Ge(0.5)
use PLOT 2 with the points shown by a +
Comment on how changing the value of p changes the shape of the distribution.
As p increases, the distribution has a larger initial value, but decays more rapidly.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.11
Discrete probability distributions
J
The Poisson probability distribution
J.1
Determining Poisson probabilities
A discrete random variable X is said to have a Poisson distribution if
λx
Pr(X=x) = x! e-λ
for x= 0,1, 2,..
The Poisson distribution is defined by a single parameter, λ.
Task:
If X~ Pn(3), evaluate
1.
Pr(X=5) 2.
Pr(X≤14)
3.
Pr(X≥2)
4.
Pr(3≤X≤9)
Poisson probabilities are evaluated through the DISTR(ibution)
menu. This is accessed by pressing 2nd [DISTR] ( VARS )
and
selecting either:
option
B:poissonpdf( to evaluate probabilities like Pr(X=x)
or
option
C:poissoncdf( to evaluate probabilities like Pr(X≤ x)
Using the calculator we see that, if X~ Pn(3):
1.
Pr(X=5) = poissonpdf(3, 5) = 0.1008 (to 4 d.p.)
2.
Pr(X≤6) = poissoncdf(3, 6) = 0.9665 (to 4 d.p.)
3.
Pr(X≥2)
= 1– Pr(x≤1)
= 1– poissoncdf(3, 1)
= 0.8009 (to 4 d.p.)
4.
Pr(3≤X≤9)
= Pr(X≤9)–Pr(X≤2)
= poissoncdf(3, 9)–poissoncdf(3, 2)
= 0.5757 (to 4 d.p.)
Exercise
1
If X~ Pn( 0.5), use your calculator to show that, correct to 4 decimal places:
(1)
(3)
Pr(X=0) = 0.6065
Pr(X≥1) = 0.3935
(2)
(4)
Pr(X≤3) = 0.9982
Pr(1≤X≤4) = 0.3933
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.12
Discrete probability distributions
J.2
Displaying Poisson probability distributions
Task:
If X~ Pn(4), display the distribution of X graphically for x= 0, 1, 2, ...10.
To display a probability distribution graphically, we need to create two lists of values,
•
the first containing the values taken by X, in this case 0, 1, 2, 3, ....10.
•
the second containing the corresponding probabilities.
Step 1.
Go to the stat list editor and create a list named X. Enter the
values 0, 1, 2,...10 into this list.*
*If you already have a list called X, either
1. simply clear it by moving the cursor onto the X at the top of the column
and press CLEAR and re-enter the required values,
or
2. edit it by deleting unwanted items or adding extra items.
Step 2.
Next, create a list named PROB which will contain the Poisson
probabilities for x= 0,1, 2....10.
*If you already have a list called PROB, simply clear it by moving the
cursor onto the X at the top of the column and press CLEAR and
proceed as follows.
Return to the HOME screen ( 2nd [QUIT] ) and enter the
following
poissonpdf(4,L X) ¿ L PROB
and press ENTER .*
*This command enters the required Poisson probabilities into the list
called PROB. If you return to the stat list editor STAT
ENTER , you
will see that the required probabilities have been entered into the list
named PROB.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.13
Discrete probability distributions
Step 3.
Go to the STATPLOT menu and set it up to display a scatterplot
with the probabilities on the vertical axis ( y-axis) and x-values on
the horizontal axis ( x-axis). Display by pressing ZOOM
9 *.
*If you are unsure how to do this, see section E.1.
If you have followed this procedure properly, you should have the
probability distribution as shown. Individual probabilities can be
read off the graph by pressing TRACE and using the horizontal
arrow keys to move from point to point.
Exercises
1.
If Y~ Pn(1.5) display the distribution of Y graphically for x = 0,1, 2, .....8.
2.
Simultaneously plot the distributions of the random variables Y1, Y2, Y3 for
x= 1, 2, ...10 where:
Y1~ Pn(2)
use PLOT 1 with the points shown by a
Y2~ Pn(5)
use PLOT 2 with the points shown by a +
Y3~ Pn(9)
use PLOT 3 with the points shown by a .
Comment on how changing the value of λ changes the shape of the distribution.
The distribution changes from being positively skewed to symmetric, while the peak moves
to the right.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors
2.14
J.3
Discrete probability distributions
The Poisson distribution as an approximation to the binomial distribution
In the limit, as n→ ∞ and p→ 0 in such a way that np→ λ, then
λx --λ
x! e
so that, for large n and small p, the binomial distribution can be approximated by the
Poisson distribution. This is easily demonstrated graphically.
nC p x(1– p)n-x →
x
Three comparisons:
Comparison 1
X~ Bi(10, 0.5) with X~ Pn(5)
for x = 0,1, 2,...10
here
λ= np =5
+ Poisson
binomial
+ Poisson
binomial
+ Poisson
binomial
Comparison 2
X~ Bi(10, 0.2) with X~ Pn(2)
for x = 0,1, 2,...10
here
λ= np =2
Comparison 3
X~ Bi(10, 0.05) with X~ Pn(0.5)
for x = 0,1, 2,...10
here
λ= np =0.5
For fixed n, the approximation clearly improves as p decreases. A
similar pattern can be seen by holding p constant and increasing n.
© Peter Jones and Chris Barling, 2001
Except for classroom use, not to be reproduced without the permission of the authors