Chapter 7 Functions of Random Variables

1
Chapter 7 Functions of Random Variables

• Wen-Hsiang Lu (???)
• Department of Computer Science and Information
  Engineering,
• National Cheng Kung University
• 2005/05/2

2
Introduction

• Moment-generating function helpful in learning
  about distributions of linear functions of random
  variables
• Statistical hypothesis testing, estimation, or
  even statistical graphics involve functions of
  one or more random variables
• The use of averages of random variables
• The distribution of sums of squares of random
  variables

3
Transformations of Variables

• Theorem 7.1 Suppose that X is a discrete random
  variable with probability distribution f(x). Let
  Y u(X) define a one-to-one transformation
  between the values of X and Y so that the
  equation y u(x) can be uniquely solved for x in
  terms of y, say x w(y). Then the probability
  distribution of Y is g(y) fw(y).
• Example 7.1 Let X is a geometric random variable
  with probability distribution
  Find the probability
  distribution of the random variable Y X2.
• solution

4
Transformations of Variables

• Theorem 7.2 Suppose that X1 and X2 are discrete
  random variables with joint probability
  distribution f(x1, x2). Let Y1 u1 (X1, X2) and
  Y2 u2(X1, X2) define a one-to-one
  transformation between the points (x1, x2) and
  (y1, y2) so that the equations y1 u1(x1, x2) ,
  y2 u2(x1, x2 ) may be uniquely solved for x1
  and x2 in terms of y1 and y2, say x1 w1(y1, y2)
  and x2 w2(y1, y2) Then the joint probability
  distribution of Y1 and Y2 is g(y1, y2)
  fw1(y1, y2), w2(y1, y2).

5
Transformations of Variables
6
Transformations of Variables

• Theorem 7.3 Suppose that X is a continuous
  random variable with probability distribution
  f(x). Let Y u(X) define a one-to-one
  transformation between the values of X and Y so
  that the equation y u(x) can be uniquely solved
  for x in terms of y, say x w(y). Then the
  probability distribution of Y is g(y)
  fw(y)J, where J w(y) and is called the
  Jacobian of the transformation.
• solution

7
Transformations of Variables

• Proof

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Transformations of Variables

• Example 7.3 Let X be a continuous random
  variable with probability distribution
• Find the probability distribution of Y 2X
  3.
• solution

9
Transformations of Variables

• Theorem 7.4 Suppose that X1 and X2 are
  continuous random variables with joint
  probability distribution f(x1, x2). Let Y1 u1
  (X1, X2) and Y2 u2(X1, X2) define a one-to-one
  transformation between the points (x1, x2) and
  (y1, y2) so that the equations y1 u1(x1, x2) ,
  y2 u2(x1, x2 ) may be uniquely solved for x1
  and x2 in terms of y1 and y2, say x1 w1(y1, y2)
  and x2 w2(y1, y2) Then the joint probability
  distribution of Y1 and Y2 is
  g(y1, y2) fw1(y1, y2), w2(y1, y2)J,where
  the Jacobian is the 2?2 determinant and
  ?x1/?y1 is simply the derivative of x1 w1(y1,
  y2) with respect to y1 with y2 held constant,
  referred to in calculus as the partial derivative
  of x1 with respect to y1. The other partial
  derivatives are defined in a similar manner.

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Transformations of Variables

• Example 7.4 Let X1 and X2 be two
  continuousrandom variables with joint
  probability distribution Find the joint
  probability distribution of Y1 X12 and Y2
  X1X2.
• Solution

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Transformations of Variables

• Theorem 7.5 Suppose that X is a continuous
  random variable with probability distribution
  f(x). Let Y u(X) define a transformation
  between the values of X and Y that is not
  one-to-one. If the interval over which X is
  defined can be partitioned into k mutually
  disjoint sets such that each of the inverse
  functions x1 w1(y), x2 w2(y),
  ... , xk wk(y) of y u(x) define a
  one-to-one correspondence, then the probability
  distribution of Y is

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Transformations of Variables

• Example 7.5 Show that Y (X - ?)2/? 2 has a
  chi-squared distribution with 1 degree of freedom
  when X has a normal distribution with mean ? and
  variance ? 2.
• Solution

13
Transformations of Variables

• Solution

14
Moments and Moment-Generating Functions

• Definition 7.1 The rth moment about the origin
  of the random variable X is given by
• The first and second moments about the origin are
  given by ?1 E(X) and ?2 E(X2), so mean ?
  ?1 and variance ? 2 ?2 - ?2.
• Definition 7.2 The moment-generating function of
  the random variable X is given by E(etX) and is
  denoted by Mx(t).

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Moments and Moment-Generating Functions

• Theorem 7.6 Let X is a random variable with
  moment-generating function Mx(t). Then
• Proof

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Moments and Moment-Generating Functions

• Example 7.6 Find the moment-generating function
  of the binomial random variable X and then use it
  to verify that ? np and ?2 npq.
• Proof

17
Moments and Moment-Generating Functions

• Example 7.7 Show that the moment-generating
  function of the random variable X having a normal
  probability distribution with mean ? and variance
  ?2 is given by
• Proof

1
18
Moments and Moment-Generating Functions

• Example 7.8 Show that the moment-generating
  function of the random variable X having a
  chi-squared distribution with v degrees of
  freedom is
• Proof

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Moments and Moment-Generating Functions

• Theorem 7.7 (Uniqueness Theorem) Let X and Y be
  two random variables with moment-generating
  functions Mx(t) and MY(t), respectively. If Mx(t)
  MY(t) for all values of t, then X and Y have
  the same probability distribution.
• Theorem 7.8
• Proof
• Theorem 7.9
• Proof

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Moments and Moment-Generating Functions

• Theorem 7.10 If X1, X2 ,, Xn are independent
  random variables with moment-generating functions
  Mx1(t), Mx2(t),, Mxn(t), respectively, and Y
  X1 X2 Xn, then
• Proof

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Moments and Moment-Generating Functions

• Example The sum of two independent random
  variables having Poisson distributions with
  parameters ?1 and ?2, has a Poisson distribution
  with parameter ?1 ?2.
• SolutionTwo independent Poisson random
  variables with moment-generating functions given
  by (Exercise 19)respectively. According to
  Theorem 7.10, Y1 X1 X2 is
• So, Y1 have a Poisson distribution with
  parameter ?1 ?2.

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Moments and Moment-Generating Functions

• Theorem 7.11 If X1, X2 ,, Xn are independent
  random variables having normal distributions with
  means ?1, ?2 ,, ?n and variances ?12, ?22 ,,
  ?n2, respectively, then the random variable
  Y a1X1 a2X2 anXnhas
  a normal distribution with mean
  ?Y a1 ?1 a2 ?2 an ?nand
  variance ?Y2
  a12?12 a22?22 an2?n2.

?Y
?Y2
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Moments and Moment-Generating Functions

• Theorem 7.12 If X1, X2 ,, Xn are mutually
  independent random variables that have,
  respectively, chi-squared distributions with v1,
  v2 ,, vn degrees of freedom, then the random
  variable Y X1
  X2 Xnhas a chi-squared distribution with
  v v1 v2 vn degrees of freedom.
• Proof

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Moments and Moment-Generating Functions

• Corollary If X1, X2 ,, Xn are independent
  random variables having identical normal
  distributions with mean ? and variances ?2
  has a chi-squared
  distribution with v n degrees of freedom.
• Example 7.5 states that each of the n independent
  random variables Y (Xi - ?) /? 2 has a
  chi-squared distribution with 1 degree of
  freedom.
• It establishes a relationship between chi-squared
  distribution and the normal distribution.
• If Z1, Z2 ,, Zn are independent standard normal
  random variables, then has a
  chi-square distribution and single parameter, v,
  the degrees of freedom, is n , the number of
  standard normal variates.

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Exercise

• 17, 19, 21