Chap 6 Continuous Random Variables Ghahramani 3rd edition

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Chap 6 Continuous Random VariablesGhahramani
3rd edition
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Outline

• 6.1 Probability density functions
• 6.2 Density function of a function of a
• random variable
• 6.3 Expectations and variances

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6.1 Probability density functions

• Def Let X be a random variable. Suppose that
  there exists a non-negative real-valued function
  f R?0, ) such that for any subset of real
  numbers A which can be constructed from intervals
  by a countable number of set operations,

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6.1 Probability density functions

• Then X is called absolutely continuous or in
  this book, for simplicity, continuous. The
  function f is called the probability density
  function, or simply, the density function of X.
• Let f be the density function of a random
  variable X with distribution function F. Some
  immediate properties of f are as follows.

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Probability density functions

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Probability density functions

• Ex 6.1 Experience has shown that while walking in
  a certain park, the time X, in minutes, between
  seeing 2 people smoking has a density function of
  the form
• (a) Calculate the value of .
• (b) Find the probability distribution function of
  X.
• (c) What is the probability that Jeff, who has
  just seen a person smoking, will see another
  person smoking in 2 to 5 minutes? In at least 7
  minutes?

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Probability density functions

• Sol

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Probability density functions

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Probability density functions

• Ex 6.2
• (a) Sketch the graph of the function
• and show that it is the probability density
• function of a random variable X.
• (b) Find F, the distribution function of X,
  and show that it is continuous.
• (c) Sketch the graph of F.
• Sol (see pp.235-237)

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6.2 Density function of a function of a random
variable

• f is the density function of a random variable X,
  then what is the density function of h(X)?
• 2 methods for it
• (a) the method of distributions to find the
  density of h(X) by calculating its distribution
  function
• (b) the method of transformations to find
  the density of h(X) directly from the density
  function of X.

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Density function of a function of a random
variable

• (Method of distributions)
• Ex 6.3 Let X be a continuous random variable with
  the probability density function
• Find the distribution and density functions of
  YX2.

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Density function of a function of a random
variable

• Sol Let G and g be the distribution function and
  the density function of Y, respectively.

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Density function of a function of a random
variable

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Density function of a function of a random
variable

• Ex 6.4 Let X be a continuous random variable with
  distribution function F and probability density
  function f. In terms of f, find the distribution
  and the density functions of YX3.
• Sol

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Density function of a function of a random
variable

• Thm 6.1 (Method of Transformations) Let X be a
  continuous random variable with density function
  fX and the set of possible values A. For the
  invertible function h A?R, let Yh(X) be a
  random variable with the set of possible values
  Bh(A)h(a) a in A. Suppose that the inverse
  of yh(x) is the function xh-1(y), which is
  differentiable for all values of y in B. Then
  fY, the density function of Y, is given by

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Density function of a function of a random
variable

• Proof Let FX and FY be distribution functions of
  X and Yh(X), respectively. Differentiability of
  h-1 implies that it is continuous. Since a
  continuous invertible function is strictly
  monotone, h-1 is either strictly increasing or
  strictly decreasing. If it is strictly
  increasing,
• and hence

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Density function of a function of a random
variable

• Moreover, in this case h is also strictly
  increasing, so
• Differentiating this by chain rule,
• which gives the theorem. If h-1 is strictly
  decreasing,
• and hence

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Density function of a function of a random
variable

• In this case h is also strictly decreasing
• Differentiating this by chain rule,
• Showing that the theorem is valid in this case as
  well.

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Density function of a function of a random
variable

• Ex 6.6 Let X be a random variable with the
  density function
• Using the method of transformation, find the
  density function of YX1/2 .

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Density function of a function of a random
variable

• Sol The set of possible values of X is A(0,
  ). Let h (0, )?R be defined by h(x)x1/2. We
  want to find the density function of Y
  h(X)X1/2. The set of possible values of h(X) is
  Bh(a) a in A(0, ). The function h is
  invertible with the inverse xh-1(y)y2, which is
  differentiable, and its derivative is
• (h-1)(y)2y. Therefore by Thm 6.1,
• .

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Density function of a function of a random
variable

• .

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Density function of a function of a random
variable

• Ex 6.7 Let X be a continuous random variable with
  the density function
• Using the method of transformation, find the
  density function of Y1-3X2 .

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Density function of a function of a random
variable

• Sol The set of possible values of X is
• A(0,1). Let h (0,1)?R be defined by
  h(x)1-3x2. The set of possible values of h(X) is
  Bh(a) a in A(-2,1). Since the domain of the
  function h is (0,1), h is invertible, and its
  inverse is found by solving 1-3x2y for x, which
  gives

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Density function of a function of a random
variable

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6.3 Expectations and variances

• Def If X is a continuous r. v. with density
  function f, the expected value of X is defined by
  

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Expectations and variances

• Ex 6.8 In a group of adult males, the difference
  between the uric acid value and 6, the standard
  value, is a random variable X with the following
  density function
• Calculate the mean of these differences for
  the group.

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Expectations and variances

• Sol
• Remark If X is a continuous r. v. with density
  function f, X is said to have a finite expected
  value if

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Expectations and variances

• Ex 6.9 A r. v. X with density function
• is called a Cauchy random variable.
• (a) Find c.
• (b) Show that EX does not exist.

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Expectations and variances

• Sol (a)
• (b) To show that EX does not exist, note that

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Expectations and variances

• Thm 6.2 For any continuous r. v. X with
  distribution function F and density function f,

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Expectations and variances

• Proof

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Expectations and variances

• Thm 6.3 Let X be a continuous r. v. with density
  function f then for any function
• h R ? R,

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Expectations and variances

• Coro Let X be a continuous r. v. with density
  function f(x). Let h1, h2, , hn be real-valued
  functions, and let a1, a2, , an be real numbers.
  Then

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Expectations and variances

• Ex 6.10 A point X is selected from the interval
  (0, ) randomly. Calculate
• E(cos2X) and E(cos2X).
• Sol

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Expectations and variances

• Def Var(X) for a continuous r. v. is similar to
  that for a discrete r. v..
• Var(X) E(X2) - (EX)2
• The moments, absolute moments, moments about a
  constant c, and central moments are all defined
  similarly.

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Expectations and variances

• Ex 6.11 The time elapsed, in minutes, between the
  placement of an order of pizza and its delivery
  is random with the density function
• (a) Determine the mean and standard deviation
  of the time it takes for the pizza shop to
  deliver pizza.
• (b) Suppose that it takes 12 minutes for the
  pizza shop to bake pizza. Determine the mean and
  standard deviation of the time it takes for the
  delivery person to deliver pizza.

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Expectations and variances

• Sol