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CHAPTER 3SOME IMPORTANT DISCRETE PROBABILITYDISTRIBUTIONS
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Introduction to Probability Distributions
Random Variable
Represents a possible numerical value from anuncertain event
Random
Variables
Discrete
Random Variable
Continuous
Random Variable
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Discrete Random Variables
Can only assume a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 comes up(then X could be 0, 1, or 2 times)
Toss a coin 5 times.Let X be the number of heads(then X = 0, 1, 2, 3, 4, or 5)
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Discrete Random VariableSummary Measures
Expected Value (or mean) of a discretedistribution (Weighted Average)
Example: Toss 2 coins,
X = # of heads,compute expected value of X: E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25)
= 1.0
X P(X)
0 0.25
1 0.50
2 0.25
===
N
1i
ii )X(PXE(X)
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Computing the Mean for Investment Returns
Return per $1,000 for two types of investments
P(XiYi) Economic condition Passive Fund XAggressive Fund Y 0.2 Recession - $ 25 - $200
0.5 Stable Economy + 50 + 60
0.3 Expanding Economy + 100 + 350
Investment
E(X) = X = (-25)(0.2) +(50)(0.5) + (100)(0.3) = 50
E(Y) = Y = (-200)(0.2) +(60)(0.5) + (350)(0.3) = 95
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Computing the Standard Deviation forInvestment Returns
P(XiYi) Economic condition Passive Fund X Aggressive Fund Y
0.2 Recession - $ 25 - $200
0.5 Stable Economy + 50 + 60
0.3 Expanding Economy + 100 + 350
Investment
43.30
(0.3)50)(100(0.5)50)(50(0.2)50)(-25 222X
=
++=
193.71
(0.3)95)(350(0.5)95)(60(0.2)95)(-200 222Y
=
++=
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The Binomial Distribution
Binomial
Hypergeometric
Poisson
ProbabilityDistributions
DiscreteProbability
Distributions
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Binomial Probability Distribution
(continued)
Observations are independent The outcome of one observation does not affect the
outcome of the other
Two sampling methods Infinite population without replacement
Finite population with replacement
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Possible Binomial Distribution Settings
A manufacturing plant labels items as eitherdefective or acceptable
A firm bidding for contracts will either get a contract
or notA marketing research firm receives survey responses
of yes I will buy or no I will not
New job applicants either accept the offer or reject it
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Example: Calculating a Binomial Probability
What is the probability of one success in fiveobservations if the probability of success is .1?
X = 1, n = 5, and p = 0.1
0.32805
.9)(5)(0.1)(0
0.1)(1(0.1)1)!(51!
5!
p)(1pX)!(nX!
n!1)P(X
4
151
XnX
=
=
=
==
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n = 5 p = 0.1
n = 5 p = 0.5
Mean
0.2
.4
.6
0 1 2 3 4 5
X
P(X)
.2
.4
.6
0 1 2 3 4 5
X
P(X)
0
Binomial Distribution
The shape of the binomial distribution depends on thevalues of p and n
Here, n = 5 and p = 0.1
Here, n = 5 and p = 0.5
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Binomial Distribution Characteristics
Mean
Variance and Standard Deviation
npE(x) ==
p)-np(12 =
p)-np(1 =Where n = sample size
p = probability of success
(1 p) = probability of failure
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n = 5 p = 0.1
n = 5 p = 0.5
Mean
0.2
.4
.6
0 1 2 3 4 5
X
P(X)
.2
.4
.6
0 1 2 3 4 5
X
P(X)
0
0.5(5)(0.1)np ===
0.67080.1)(5)(0.1)(1p)np(1-
=
==
2.5(5)(0.5)np ===
1.118
0.5)(5)(0.5)(1p)np(1-
=
==
Binomial Characteristics
Examples
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