Chap 7 Special Continuous Distributions Ghahramani 3rd edition

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Chap 7 Special Continuous DistributionsGhahraman
i 3rd edition
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Outline

• 7.1 Uniform random variable
• 7.2 Normal random variable
• 7.3 Exponential random variables
• 7.4 Gamma distribution
• 7.5 Beta distribution
• 7.6 Survival analysis and hazard function

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7.1 Uniform random variable

• Def A random variable X is said to be uniformly
  distributed over an interval
• (a, b) (written as XU(a,b) in short) if its
  density function is

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Uniform random variable

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Uniform random variable
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Uniform random variable

• Comparison
• If Y is a discrete random variable selected
• from the set 1, 2, , N , then

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Uniform random variable

• Ex 7.3 What is the probability that a random
  chord of a circle is longer than a side of an
  equilateral triangle inscribed into the circle?

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Uniform random variable
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Uniform random variable

• Sol
• (a)interpretation 1 P(dltr/2)1/2
• (b)interpretation 2 1/3
• (c)interpretation 3

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7.2 Normal random variable

• De Moivres Thm Let XB(n,1/2) then for
• a and b, a lt b
• Note that EXn/2 and s.d.(X)n1/2/2

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Normal random variable

• Thm 7.1 (De Moivre-Laplace Thm)
• Let XB(n,p) then for a and b, a lt b
• Note that EXnp and s.d.(X)(np(1-p))1/2

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Normal random variable

• Def A random variable X is called standard normal
  (written as XN(0,1)) if its distribution
  function is

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Normal random variable

• To prove is a
  distribution function

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Normal random variable

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Normal random variable

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Normal random variable

• By the fundamental theorem of calculus, the
  density function f is
• which is a bell-shaped curve that is
  symmetric about the y-axis

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Normal random variable

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Normal random variable

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Normal random variable

• Correction for continuity

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Normal random variable

• Histogram of X and the density function f

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Normal random variable

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Normal random variable

• Ex 7.4 Suppose that of all the clouds that are
  seeded with silver iodide, 58 show splendid
  growth. If 60 clouds are seeded with silver
  iodide, what is the probability that exactly 35
  show splendid growth?

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Normal random variable

• Sol

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Normal random variable

• Continue

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Normal random variable

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Normal random variable

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Normal random variable

• Def A random variable X is called normal, with
  parameters and (written as XN( , )),
  if its density function is

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Normal random variable

• Lemma If XN( , ), then Z(X- )/ is
  N(0,1). That is , if X N( , ), the
  standardized X is N(0,1).

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Normal random variable

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Normal random variable

• Ex 7.5 Suppose that a Scottish soldiers chest
  size is normally distributed with mean 39.8 and
  standard deviation 2.05 inches, respectively.
  What is the probability that of 20 randomly
  selected Scottish soldiers, 5 have a chest of at
  least 40 inches?

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Normal random variable

• Sol

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Normal random variable

• Ex 7.7 The scores on an achievement test given to
  100,000 students are normally distributed with
  mean 500 and standard deviation 100. What should
  the score of a student be to place him among the
  to 10 of all students?

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Normal random variable

• Sol to find x such that P(Xltx)0.90.

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7.3 Exponential random variable

• Def A continuous random variable X is called
  exponential with parameter gt0 (written as
  XEP( )) if its density function and
  distribution function are

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Exponential random variable

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Exponential random variable

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Exponential random variable

• Examples
• The interarrival time between 2 customers at a
  post office.
• The duration of Jims next telephone call.
• The time between 2 consecutive earthquakes in
  California.
• The time between two accidents at an
  intersection.
• The time until the next baby is born in a
  hospital.
• The time until the next crime in a certain town.

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Exponential random variable

• Ex 7.10 Suppose that every 3 months, on average,
  an earthquake occurs in California. What is the
  probability that the next earthquake occurs after
  3 but before 7 months?

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Exponential random variable

• Sol

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Exponential random variable

• Ex 7.11 At an intersection, there are 2 accidents
  per day, on average. What is the probability
  that after the next accident there will be no
  accidents at all for the next 2 days?

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Exponential random variable

• Sol

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Exponential random variable

• An important feature of exponential distribution
  is its memoryless property.

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Exponential random variable

• Exponential random variables are memoryless.
• ltproofgt

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Exponential random variable

• Ex 7.12 The lifetime of a TV tube (in years) is
  an exponential random variable with mean 10. If
  Jim bought his TV set 10 years ago, what is the
  probability that its tube will last another 10
  years?
• Sol

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• Skip
• 7.4 Gamma distribution and
• 7.5 Beta distribution
• 7.6 Survival analysis and hazard functions