Lecture 10: Continuous RV

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Lecture 10 Continuous RV
Everything existing in the universe is the fruit
of chance. Democritus

• Probability Theory and Applications
• Fall 2005
• September 29

September 29
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To Come

• More on MGF after we learn continuous
  distributions.
• Test 10/2
• Covers materials up through this lecture
• Bring Calculator
• Bring one page of notes (both sides fine)
• Sample exam on web (sorry no answers)

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WARNING

• To properly specify a CDF you give it for all
  possible values.

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RIGHT
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WRONG WRONG WRONG

• WRONG
• WRONG
• WRONG

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Outline

• Motivating Example for CRV
• Continuous Random Variables
• Sample types of problems

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Imagine.

• Driving down on a 10 mile stretch of highway near
  Roswell New Mexico.

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Suddenly

• a UFO appears

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The spaceship

• bathes you in bright light.

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Then

• the spaceship, you and your car disappears.

?
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Alien Abduction Problem

• Imagine Fox and Mulder are driving down a 10 mile
  stretch of highway and they will be abducted by
  aliens stretch of highway.
• What is the probability they will be abducted in
  the first 10 miles assuming that their chance of
  getting abducted as any point of the road is
  equally likely?

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Discrete Version

• There are mile markers that divide highway into
  10 segments.
• Let X1,2,..,10 be the probability you vanish
  after x-1 and up to mile marker x.
• X is discrete uniform.
• P(Xx)1/10 x1,..,10
• Note and sketch CDF

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Continuous Version

• Let X needs to be a real random variable since we
  could disappear at about 3.2 miles and that is
  different than 3.9 miles.
• Uniform assumption

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0
0
10
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Looks good

• Probability they disappear in the first half
• P(X5)5/101/2
• Seem like right cdf. What would the pdf be?

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Probability of small interval

• Probably disappear between point a and b
  F(b)-F(a).
• Probability disappear on a very small interval.
• Let
• by fundamental theorem of calculus

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Alien Abduction PDF

• Differentiate CDF to get PDF

.1
0
10
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Continuous RV

• X is a continuous R.V., if and only if
• F(x)P(Xx) is a continuous function from
• the reals to 0,1
• If F(x) is an integral of some function f(x)0
  of the form then f(x)
  is called a probability density function p.d.f

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Back to example

• Cdf
• pdf

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PDF

• If F(X) is the cdf of a random variable and F(x)
  is an integral of some function
• of the form
• then is called the probability density
  function (pdf)

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Recall

• Fundamental theorem of Calculus

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Alien Abudction

• CDF
• PDF

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Probability of Event

• Let X be a continuous R.V. with cdf F(x) and pdf
  f(x).
• Let A be an event (subset of R).

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Alien Abduction

• Probability abducted in 1.3 to 2.4 miles

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Alien Abduction

• Probability abducted at 1.3 miles
• The probability Xx for any x is 0!!

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Note

• Let X be a R.V. with pdf f(x)

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Problem Type I

• Given that x has pdf
• Find c

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continued

• pdf of X is

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Problem Type 2

• Find cdf of X for previous problem

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Problem Type 3

• Find P(1/4
• Using cdf F(4)-F(1/2)1-(3/4-1/4)1/2
• Using pdf

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Problem Type 3

• Find P(X1/3X1/2)