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Lecture 8

Lecture 8 (week 9)

10 Pages
161 Views

Department
Quantitative Methods
Course Code
QMS 102
Professor
Clare Chua

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1
Topics that will be tested in
Quiz 2:
Discrete Probability
Binomial Probability
Poisson Probability
Normal Probability
Key Answers
Page 241
5.10) E(X)=$80,000
5.11a)
# years XP(X)
1$2.00 0.5
23.50 0.2
34.50 0.2
56.50 0.1
5.11b) $406.25
5.12a) Project 3
b) Project 3
5.13) Country C
Q5.14) Tire C
Q 5.15) Order 11 units
Binomial Probability Distribution
Which of the above probability
distributions are for a discrete
random variable? Continuous?
Discrete Continuous
Binomial Normal
Poisson Inverse
Normal
Central Limit
Theorem
Discrete
Random Variables
A discrete random variablediscrete random variable may assume either a finite number
of values or an infinite sequence of values.
FINITE
Number of values
INFINITE
Number of values
How to differentiate between
discrete and continuous random
variables?
DISCRETE
The observation can be
counted
CONTINUOUS
The observation can be
measured
ElTi(i
Example: the number of
students who drive to
school
What is the probability
that at most 110 students
will drive to school?
E
xamp
l
e:
Ti
me
(i
n
seconds) is measured and
not counted.
What is the probability
that the download time for
a home page on a Web
browser is between 7 and
10 seconds?
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2
Discrete Random Variable
Examples: Finite/Infinite number
values
Experiment Random
Variable
Possible
Values
Make 100 sales calls
#Sales
0,1,2, ..., 100
Fi it
Make
100
sales
calls
#
Sales
0,
1,
2,
...,
100
Inspect 70 radios # Defective 0, 1, 2, ..., 70
Answer 33 questions # Correct0, 1, 2, ..., 33
Count cars at toll
between 11:00 & 1:00
# Cars
arriving
0, 1, 2, ..., f
Fi
n
it
e
Number
values
Infinite
Number
values
What is a Binomial Probability
Distribution?
Characteristics of the Binomial
Distribution
1. The experiment consists of a sequence of n
identical trials (repetitions).
2. Each trial has only one of the two possible mutually
exclusive outcomes, success or a failure. Two
outcomes, success and failure, are possible on each
trial.
3. The probability of each outcome does not change
from trial to trial.
4. The trials are independent, thus we must sample with
replacement.
The random variable that we are interested in is X,
Number of successes’ in a sample
of nobservations (trials)
X= # of successes
Summary: Three pieces of info
1. Sequence of nidentical trials / A sample of ‘n
items are selected from a large population
2. Constant Probability (3) for each Trial
e.g. Probability of getting atail is the sameeach
e.g.
Probability
of
getting
a
tail
is
the
same
each
time we toss the coin and each light bulb has the
same probability of being defective
3. Each trial has 2 outcomes
Success’ (desired outcome) or ‘failure’
X= # of successes
To calculate the probability that X takes on a
specific value we use the Binomial Probability
Distribution Function
What is the Binomial Probability
Distribution Function?
xnx
xnx
n
xXP
)1(
)!(!
!
)(
SS
P(X=x) = probability that x successes given a knowledge of
n and 3
X= number of ‘successes in
sample, (X= 0, 1, 2, ..., n)
S= probability of each ‘success
n= sample size
Use the CASIO calculator to obtain the
Binomial Probability
See Handout on “CASIO Calculator Lesson 3 :
Binomial and Poisson – page 250
STAT F5(DIST) F5(BINM) F1(Bpd)
Using the CASIO calculator
Then select the following options
You will see: Binomial P.D
Data : Variable
X :
Numtrial :
P :
Save Res: None
Execute
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3
Bpd and Bcd functions
The Casio calculator has 2 Binomial Probability
functions programmed into its memory.
These are:
1) Bpd stands for Binomial Probability Distribution.
This function calculators a Binomial Probability of the form:
P( X#)
P(
X
=
#)
2) Bcd stands for Binomial Cumulative Distribution.
This function calculators a Binomial Probability of the form:
P( X #)
Use the CASIO calculator to obtain the
Binomial Probability
See Handout on “CASIO Calculator
Lesson 3 : Binomial and Poisson – page 250
X
=
# oftails
Experiment: Toss 1 coin 6 times in a row..
What’s the probability of getting 2 tails?
X
#
of
tails
n = 6
3= 0.5
Then select the following options
You will see:
Binomial P.D
Data : Variable
x : 0
Numtrial : 0
P : 0
Save Res: None
Execute
2 EXE
6 EXE
0.5 EXE
F2 (CALC)
Presentation of your Solution
(for Binomial Probability)
BINOMIAL TEMPLATE
Your answers should look like the following template:
X = # of _____________________
(successes)
n = __________________
= _________________
(decimals)
P ( ) = P ( X symbol # ) = Bpd ( ) = 0.xxxx
(words) ( <, >, , , =) or Bcd ( ) 4 decimals
If the question does not directly say what the number is.
Example 5.11
A survey of people in the 30 and under 40 age bracket shows
that 43% of them have investments in mutual funds. In a
particular condominium there are 18 adults in this age
bracket. What is the probability that:
1. 5 of them will have investments in
mutual funds?
mutual
funds?
2.At most 2 of these people will have investments in mutual
funds?
3.More than 10 of these people will have investments in
mutual funds?
4.From 3 to 9 of these people will have investments in mutual
funds?
1) What is the probability that 5 of them will
have investments in mutual funds?
How do you write the probability?
P(X=5)
Define X:
What is X? What are you observing or counting???
X= # of people in their 30’s with investments in mutual
funds.
Next, state n and
Where do you obtain the values of n and ?
A survey of people in the 30 and under 40 age bracket shows
that 43% of them have investments in mutual funds.
In a particular condominium there are 18 adults in this age bracket.
1) What is the probability that 5 of them will
have investments in mutual funds?
X= # of people in their 30’s with investments in mutual
funds.
n
=
18
n
18
= 0.43 = 43%WRONG!
P(X=5) = Bpd (5, 18, 0.43) = 0.0844
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Description
Key Answers Topics that will be tested in Page 241 Quiz 2: 5.10) E(X)=$80,000 5.11a) Discrete Probability # years X P(X) 1 $2.00 0.5 Binomial Probability 2 3.50 0.2 Poisson Probability 3 4.50 0.2 5 6.50 0.1 Normal Probability 5.11b) $406.25 Q5.14) Tire C 5.12a) Project 3 b) Project 3 Q 5.15) Order 11 units 5.13) Country C Which of the above probability distributions are for a discrete random variable? Continuous? Discrete Continuous Binomial Probability Distribution Binomial Normal Poisson Inverse Normal Central Limit Theorem Addicreterandom variabe may assume either a finite number How to differentiate between of values or an infinite sequence of values. discrete and continuous random Discrete variables? CONTINUOUS Random Variables DISCRETE The observation can be The observation can be counted measured Example: the number of EExamplleTiime ((iin students who drive to seconds) is measured and FINITE INFINITE school not counted. Number of values Number of values What is the probability What is the probability that at most 110 students that the download time for will drive to school? a home page on a Web browser is between 7 and 10 seconds? 1 www.notesolution.com Discrete Random Variable Examples: Finite/Infinite number values Experiment Random Possible What is a Binomial Probability Variable Value s Maake 1100sales calslls# Saless 0,1,,2,, .1000.,, 1 Distribution? Fiiniitte Inspect 70 radios # Defective 0, 1, 2, ..., 70 Number values Answer 33 questions # Correct 0, 1, 2, ..., 33 Count cars at toll # Cars 0, 1, 2, ..., B between 11:00 & 1:00 arrivin g Infinite Number values Characteristics of the Binomial Distribution Summary: Three pieces of info 1. The experiment consists of a sequence of n identical trials (repetitions). 2. Each trial has only one of the two possible mutually 1. Sequence of n identical trials / A sample of n exclusive outcomes, success or a failure. Two items are selected from a large population outcomes, success and failure, are possible on each 2. Constant Probability (!) for each Trial trial. eg..Probabiiy ofgettnga aiillhee ame each time we toss the coin and each light bulb has the 3. The probability of each outcome does not change same probability of being defective from trial to trial. 4. The trials are independent, thus we must sample with 3. Each trial has 2 outcomes replacement. Success (desired outcome) or failure The random variable that we are interested in is, X= # of successes To calculate the probability that X takes on a Number of successesin a sample specific value we use the Binomial Probability of n observations (trials) Distribution Function X= # of successes Use the CASIO calculator to obtain the What is the Binomial Probability Binomial Probability See Handout on CASIO Calculator Lesson 3 : Distribution Function? Binomial and Poisson page 250 n! x n x Using the CASIO calculator P ( x) 5 (1 ) x!( )! STAT F5(DIST) F5(BINM) F1(Bpd) P(X=x) = probability that x successes given a knowledge of Then select the following options n and ! You will see: Binomial P.D X = number of successes in Data : Variable sample, (X = 0, 1, 2, ..., n) X : Numtrial : 5 = probability of each success P : n = sample size Save Res: None Execute 2 www.notesolution.comBpd and Bcd functions Use the CASIO calculator to obtain the The Casio calculator has 2 Binomial Probability Binomial Probability functions programmed into its memory. Experiment: Toss 1 coin 6 times in a row.. These are: Whats the probability of getting 2 tails? See Handout on CASIO Calculator 1) Bpd stands for Binomial Probability Distribution. This function calculators a Binomial Probability of the form: Lesson 3 : Binomial and Poisson page 250 X= #offaiillssTTAT FF5(DIST))F5(BINM)) F1(Bppd)) P((X= #)) n = 6 Then select the following options
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