(c) 2007 iupui spea k300 (4392) outline random variables expected values data generation process...
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(c) 2007 IUPUI SPEA K300 (4392)
Outline
Random VariablesExpected ValuesData Generation Process (DGP)Uniform Probability DistributionBinomial Probability Distribution
(c) 2007 IUPUI SPEA K300 (4392)
Random Variables
Variables whose values change randomly Neither predictable, nor constant Not arbitrary, but stochastic (followed by a
data generation process) Statistically independent P(x=v): probability that X takes a particular
value of v. Examples
The number of heads when tossing a coin The number you get when rolling a die
(c) 2007 IUPUI SPEA K300 (4392)
Discrete versus Continuous
Discrete if the set of outcomes is either finite in number or countably infiniteExample: tossing a coin, rolling a die, the
number of customers arrived per hourContinuous if the set of outcomes is
infinitively divisible and not countable. Example: GPA, height, gas mileage
(c) 2007 IUPUI SPEA K300 (4392)
Level of Measurement
Rank Difference Ratio Continuous Operator
Nominal N/A N/A N/A Discrete =
Ordinal Yes N/A N/A Discrete =, >, <
Interval Yes Yes N/A Yes =, >, <, +, - , x , ÷
Ratio Yes Yes Yes Yes =, >, <, +, - , x , ÷
Count Yes Yes Yes Discrete =, >, <, +, -, x, ÷
(c) 2007 IUPUI SPEA K300 (4392)
Rules of Probability
0<=P(X=v)<=1∑P(X=v)=1P(X=v)=0 when never happeningP(X=v)=1 when always happening
(c) 2007 IUPUI SPEA K300 (4392)
Uniform Distribution
Each event has the same probability Examples
Tossing a coil: 1/2 rolling a die: 1/6
X 1 2 3 4 5 6
P(X) 1/6 1/6 1/6 1/6 1/6 1/6
(c) 2007 IUPUI SPEA K300 (4392)
Expected Value 1
Expected value of a probability distribution is equivalent to mean
∑xP(x)=x1*P(x1)+ x2*P(x2)+…
Expected value of rolling a die: 1*1/6 + 2*1/6 + 3*1/6 + 4*1/6 + 5*1/6 +6*1/6 = 21/6=3.5
(c) 2007 IUPUI SPEA K300 (4392)
Expected Value 2
Let us play a game of tossing a coin. You have 50% chance to get a head.
If you get a head, you will get $500; otherwise (getting a tail) you will lose $400.
Expected value:+$500*1/2 + (-$400)*1/2 =$50
Do you want to play this game?
(c) 2007 IUPUI SPEA K300 (4392)
Variance of Probability Distribution
σ2 = ∑[(x-µ)2P(x)] σ2 = ∑x2P(x)-µ2 Example 5-9. µ=3.5σ2 = (1-3.5)2*1/6 +(2-3.5)2*1/6 …=2.9σ2 = 12*1/6 +22*1/6 … -(3.5)2=2.9
(c) 2007 IUPUI SPEA K300 (4392)
Data Generation Process (DGP)
How are data (random variables) generated?
“Tossing a coin” is a DGP that has only two outcomes (head/tail) with equal P
“Rolling a die” is a DGP that has six outcomes with equal P (1/6)
How about GPA of SPEA students?How do we formulate these DGPs
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Experiment
Each trial: Bernoulli process (or trial)There must be a fixed number of trials, nEach trial can have only two outcomes, x
(yes/no, true/false, success/failure)The outcome of each trial must be
(statistically) independent of each otherThe probability of a success, p, must
remain the same for each trial
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Distribution 1
Consists of N times of the Bernoulli trial with two outcomes
Probabilities of these outcomes, which are not necessarily equal
Examples: What is the probability that you will get 10 heads
when tossing a coin 20 times? Given .01 of getting a certain disease, what is the
probability that 5 out of 10 randomly selected subjects get the disease?
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Distribution 2
Tossing a coin four times (16 outcomes=2^4) X is the number of times to get the head (0
through four) P is the probability of getting the head
4C0= 4C4 =1, 4C1 = 4C3=4
4C2 =4!/[2!(4-2)!]=4!/(2!2!)=6
X 0 1 2 3 4 Total
P(X) 1/16 4/16 6/16 4/16 1/16 1
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Distribution 3
P(p) probability of successP(q) probability of failure, P(q)=1-P(p)N the number of trialsX the number of successes in n trials
)()(
)!(!
!)( xnxxnx
xn qpxnx
nqpCxP
16
6
2
16
2
1
2
1
)!24(!2
!4)(
4242)24(2
24
qpCxP
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Distribution 4
What is the probability that you will get 10 heads when tossing a coin 20 times?
Given .01 of getting a certain disease, what is the probability that 5 out of 10 randomly selected subjects get the disease?
)510(5)510(5510 99.01.
)!510(!5
!10)(
qpCxP
)1020(10)1020(10
1020 2
1
2
1
)!1020(!10
!20)(
qpCxP
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Distribution 5
Sick and tired of the formula? Good news. Someone computed the
probability of binomial distribution for you. See example 5-18, p 265.
What you have to know is N, X, and PGo to page 626What is the probability that 5 out of 10
get the disease? P=.01
(c) 2007 IUPUI SPEA K300 (4392)
Binomial Distribution 6
Expected value: µ=npVariance: npqStandard deviation: sqrt(npq)Example 5.21, p. 267.
µ=np=4 X ½= 2σ2 =npq=4X½ X½ =1σ=sqrt(1)=1
(c) 2007 IUPUI SPEA K300 (4392)
Illustration 0
Tossing a coinTwo outcomes: head and tailP(head)=1/2http://www.uwsp.edu/psych/stat/9/
hyptestd.htm
(c) 2007 IUPUI SPEA K300 (4392)
Illustration 1, N=1 and 20
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0 1Random Variable X
Binomial Probability (N=1, p=1/2)
0.2
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0 1 2Random Variable X
Binomial Probability (N=2, p=1/2)
0.2
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0 1Random Variable X
Binomial Probability (N=1, p=1/10)
0.2
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0 1 2Random Variable X
Binomial Probability (N=2, p=1/10)
(c) 2007 IUPUI SPEA K300 (4392)
Illustration 2, N=3 and 40
.2.4
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0 1 2 3Random Variable X
Binomial Probability (N=3, p=1/2)
0.2
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0 1 2 3 4Random Variable X
Binomial Probability (N=4, p=1/2)
0.2
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0 1 2 3Random Variable X
Binomial Probability (N=3, p=1/10)
0.2
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0 1 2 3 4Random Variable X
Binomial Probability (N=4, p=1/10)
(c) 2007 IUPUI SPEA K300 (4392)
Illustration 3, N=5 and 100
.2.4
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0 1 2 3 4 5 6 7 8 9 10Random Variable X
Binomial Probability (N=10, p=1/2)
0.2
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0 1 2 3 4 5Random Variable X
Binomial Probability (N=5, p=1/2)
0.2
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0 1 2 3 4 5Random Variable X
Binomial Probability (N=5, p=1/10)
0.2
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0 1 2 3 4 5 6 7 8 9 10Random Variable X
Binomial Probability (N=10, p=1/10)
(c) 2007 IUPUI SPEA K300 (4392)
Illustration 4, N=50 and 100
0.0
3.0
6.0
9.1
2.1
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0 20 40 60 80 100Random Variable X
Binomial Probability (N=100, p=1/10)
0.0
5.1
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0 10 20 30 40 50Random Variable X
Binomial Probability (N=50, p=1/10)
0.0
3.0
6.0
9.1
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0 20 40 60 80 100Random Variable X
Binomial Probability (N=100, p=1/2)
0.0
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0 10 20 30 40 50Random Variable X
Binomial Probability (N=50, p=1/2)