04 Discrete and Continuous Random Variables

8/2/2019 04 Discrete and Continuous Random Variables

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Random Variables

Discrete and Continuous

Friday, April 20, 2012 1Dr. S. Jain


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Data Types

Data

Numerical Qualitative

Discrete Continuous



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7.3

Random Variables

A random variable is a function or rule that assigns anumber to each outcome of an experiment.Basically itis just a symbol that represents the outcome of anexperiment.

X = number of heads when the experiment is flipping acoin 20 times.

C = the daily change in a stock price.

R = the number of miles per gallon you get on your

auto during a family vacation. Y = the amount of medication in a blood pressure pill.

V = the speed of an auto registered on a radar detectorused on I-20

Friday, April 20, 2012 Dr. S. Jain


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Types of

Random Variables

Discrete Random Variable

Whole Number (0, 1, 2, 3 etc.)

Countable, Finite Number of Values

Jump from one value to the next and cannot take any values in

between.

Continuous Random Variables

Whole or Fractional Number

Obtained by Measuring

Infinite Number of Values in Interval

Too Many to List Like Discrete Variable



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7.5

Types of Random Variables

Discrete Random Variable usually count data [Number of]

one that takes on a countable number of values this means you can sit downand list all possible outcomes without missing any, although it might take youan infinite amount of time.

X = values on the roll of two dice: X has to be either 2, 3, 4, , or 12.

Y = number of accidents on the XYZ campus during a week: Y has to be 0, 1, 2,3, 4, 5, 6, 7, 8, real big number

Continuous Random Variable usually measurement data [time, weight,distance, etc]

one that takes on an uncountable number of values this means you cannever list all possible outcomes even if you had an infinite amount of time.

X = time it takes you to drive home from class: X > 0, might be 30.1 minutesmeasured to the nearest tenth but in reality the actual time is30.10000001. minutes?)

Exercise: try to list all possible numbers between 0 and 1.

Friday, April 20, 2012 Dr. S. Jain


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Discrete random variable

A discrete random variable is one which may take on only acountable number of distinct values such as 0, 1, 2, 3, 4, ...

Discrete random variables are usually (but not necessarily)

counts. If a random variable can take only a finite number ofdistinct values, then it must be discrete.

Examples of discrete random variables include the number ofchildren in a family, the Friday night attendance at a cinema,

the number of patients in a doctor's surgery, the number ofdefective light bulbs in a box of ten.



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Discrete Random Variable

Examples

Experiment Random

Variable

Possible

Values

Children of One Gender

in Family

# Girls 0, 1, 2, ..., 10?

Answer 33 Questions # Correct 0, 1, 2, ..., 33

Count Cars at Toll

Between 11:00 & 1:00

# Cars

Arriving0, 1, 2, ...,

Open Check in Lines # Open 0, 1, 2, ..., 8


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Discrete

Probability Distribution

1. List of All possible [x, p(x)] pairs

x= Value of Random Variable (Outcome)

p(x) = Probability Associated with Value

2. Mutually Exclusive (No Overlap)

3. Collectively Exhaustive (Nothing Left Out)

4. 0 p(x) 1

5. p(x) = 1



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Marilyn says: It may sound strange, but more families of 4 children

have 3 of one gender and one of the other than any other

combination. Explain this.

Construct a sample space and look at the total number of ways

each event can occur out of the total number of combinations

that can occur, and calculate frequencies.

Sample Space

BBBB

GBBB

BGBB

BBGB

BBBG

GGBB

GBGB

GBBG

BGGB

BGBG

BBGG

BGGG

GBGG

GGBG

GGGBGGGG

P (girl) = 1/2P (boy) = 1/2

so, P (BBBB) = x x x = 1/16

Are all 16 combinations equally likely? Is the sex ofeach child independent of the other three?

If you have a family of four, what is the probability of

P(all girls or all boys) =

P (2 boys, 2 girls)=

P(3 boys, 1 girl or 3 girls, 2 boy)=8/16=4/8=1/2 8 ways to have 3 of 1 and 2 ofthe other.

6/16 = 3/8 six different ways to have 2 boys and 2 girls

2/16 = 1/8



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What is the probability of exactly 3 girls in 4 kids?

What is the probability of at least 3 girls in 4 kids?

Assume the random variable X represents the number of girls in a

family of 4 kids. (lower case x is a particular value of X, ie: x=3 girls in

the family)

P(X=3) = 4/16

Sample Space

BBBB

GBBB

BGBB

BBGB

BBBG

GGBB

GBGB

GBBG

BGGB

BGBG

BBGG

BGGG

GBGG

GGBG

GGGBGGGG

Random Variable X

x=0

x=1

x=1

x=1

x=1

x=2

x=2

x=2

x=2

x=2x=2

x=3

x=3

x=3

x=3

x=4

Number ofGirls, x

Probability,P(x)

0 1/16

1 4/16

2 6/16

3 4/16

4 1/16

Total 16/16=1.00

P(X3) = 5/16Friday, April 20, 2012 10Dr. S. Jain


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Visualizing Discrete Probability

Distributions

Probability, P(x)

1/16

6/16

4/16 4/16

1/16

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 1 2 3 4

Number of Girls, x

P(x)

Listing TableNumber of Girls, x Probability, P(x)

0 1/16

1 4/16

2 6/16

3 4/16

4 1/16

Total 16/16=1.00

{(0,1/16), (1,.25), (2,3/8),(3,.25),(4,1/16) }

Graph

X is random and x is fixed. We can

calculate the probability that

different values of X will occur and

make a probability distribution.



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Probability Distributions

Probability, P(x)

1/16

6/16

4/16 4/16

1/16

0.00

0.05

0.10

0.15

0.200.25

0.30

0.35

0.40

0 1 2 3 4

Number of Girls, x

P(x)

Probability distributionscan be written as probability histograms.

Cumulative probabilities: Adding up probabilities of a range of values.


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Discrete example: roll of a die

x

p(x)

1/6

1 4 5 62 3

xall

1P(x)



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Probability mass function (pmf)

x p(x)

1 p(x=1)=1/6

2 p(x=2)=1/6

3 p(x=3)=1/6

4 p(x=4)=1/6

5 p(x=5)=1/6

6 p(x=6)=1/6

1.0Friday, April 20, 2012 14Dr. S. Jain


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Cumulative distribution function

(CDF)

x

P(x)

1/6

1 4 5 62 3

1/3

1/2

2/3

5/6

1.0



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Cumulative distribution function

x P(xA)

1 P(x1)=1/6

2 P(x2)=2/6

3 P(x3)=3/6

4 P(x4)=4/6

5 P(x5)=5/6

6 P(x6)=6/6



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Data Types

Data

Numerical Qualitative

Discrete Continuous



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Continuous Random Variable

A variable with many possible values at all

intervals



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Examples

Experiment Random

Variable

Possible

Values

Weigh 100 People Weight 45.1, 78, ...

Measure Part Life Hours 900, 875.9, ...

Ask Food Spending Spending 54.12, 42, ...

Measure Time

Between Arrivals

Inter-Arrival

Time

0, 1.3, 2.78, ...


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Continuous Probability Density

Function

1. Mathematical Formula

2. Shows All Values, x, &

Frequencies, f(x) f(X) Is NotProbability

3. Properties

Area under curve sums to 1

Can add up areas of function to

get probability less than a

specific value Value

(Value, Frequency)

Frequency

f(x)

a bx


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Probability

Probability Is Area

Under Curve!

1984-1994 T/Maker Co.

P c x d( )

f(x)

Xc d


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Consider the following table of sales,

divided into intervals of 1000 units each,

interval

(0,1000]

(1000,2000](2000,3000]

(3000,4000]

(4000,5000]

(5000,6000]

(6000,7000]



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and the relative frequency of each interval.

interval relativefreq.

(0,1000] 0

(1000,2000] 0.05

(2000,3000] 0.25

(3000,4000] 0.30

(4000,5000] 0.25

(5000,6000] 0.10

(6000,7000] 0.05

1.00Friday, April 20, 2012 23Dr. S. Jain


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Were going to divide the relative frequencies

by the width of the cells (which here is 1000).

This will make the graph have an area of 1.

intervalrelative

freq.

(0,1000] 0 0

(1000,2000] 0.05 0.00005

(2000,3000] 0.25 0.00025

(3000,4000] 0.30 0.00030

(4000,5000] 0.25 0.00025

(5000,6000] 0.10 0.00010

(6000,7000] 0.05 0.00005

widthcell

freq.relativef(x)



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Graph

interval

(0,1000] 0

(1000,2000] 0.00005

(2000,3000] 0.00025(3000,4000] 0.00030

(4000,5000] 0.00025

(5000,6000] 0.00010

(6000,7000] 0.00005

widthcell

freq.relativef(x)

0 1000 2000 3000 4000 5000 6000 7000

sales

f(x) = p(x)

0.00030

0.00025

0.00020

0.00015

0.00010

0.00005

0

The area of each bar is the frequency of the category, so the

total area is 1.Friday, April 20, 2012 25Dr. S. Jain


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Graph

interval

(0,1000] 0

(1000,2000] 0.00005

(2000,3000] 0.00025(3000,4000] 0.00030

(4000,5000] 0.00025

(5000,6000] 0.00010

(6000,7000] 0.00005

widthcell

freq.relativef(x)

0 1000 2000 3000 4000 5000 6000 7000

sales

f(x) = p(x)

0.00030

0.00025

0.00020

0.00015

0.00010

0.00005

0

Here is the frequency polygon.Friday, April 20, 2012 26Dr. S. Jain


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If we make the intervals 500 units instead of 1000, the

graph would probably look something like this:

sales

f(x) = p(x)

The height of thebars increases and

decreases more

gradually.



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If we made the intervals infinitesimally small, the bars

and the frequency polygon would become smooth,

looking something like this:

f(x) = p(x)

sales

This what the distribution

of a continuous random

variable looks like.

This curve is denoted f(x) or

p(x) and is called the

probability density

function.



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pmf versus pdf

For a discrete random variable, we had aprobability mass function (pmf).

The pmf looked like a bunch of spikes, and

probabilities were represented by the heightsof the spikes.

For a continuous random variable, we have a

probability density function (pdf). The pdf looks like a curve, and probabilities

are represented by areas under the curve.


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04 Discrete and Continuous Random Variables