010'141 Engineering Mathematics II Lecture 7 Joint Probability Functions

1
010.141 Engineering Mathematics IILecture
7Joint Probability Functions

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University
• Partly based on
• Sheldon Ross A First Course in Probability

2
Outline

• Pairwise Joint Distribution Functions
• Example the Rayleigh Distribution

3
Joint Distribution Functions

• It commonly arises that
• We know the joint distribution of continuous
  random variables X1 and X2
• We have two functionally dependent distributions
  Y1 g1(X1,X2) and Y2 g2(X1,X2)?
• We would like to know the joint distribution of
  Y1 and Y2
• Can we give a general formula for this joint
  distribution?
• In fact, we require some extra constraints on Y1
  and Y2

4
Joint Distribution Functions

• Under these conditions, Y1 and Y2 are jointly
  continuous
• Their joint probability density function is
• fY1,Y2(y1,y2) fX1,X2(x1,x2) / J (x1,x2)

5
Constraints onJoint Distribution Functions

• The relationship between (X1,X2) and (Y1,Y2) must
  be an equivalence relation
• That is, there exist h1and h2 such that ify1
  g1(x1,x2) and y2 g2(x1,x2) thenx1 h1(y1,y2)
  and x2 h2(y1,y2)
• The Jacobean Determinant J (x1,x2) ?g1/?x1
  ?g1/?x2 ?g2/?x1 ?g2/?x2 is nonzero

6
Joint Distribution Functions Example

• In mechanics, we often wish to switch from
  Cartesian to Polar coordinates
• Suppose X and Y are normally distributed (with
  equal means and standard deviations) and
  independent
• For example, we might be shooting arrows at a
  target, with our x and y errors being normally
  distributed, and independent of each other
• We can use the transformation
• r ? (x2 y2)?
• ? tan-1(y/x)?
• What is the joint distribution of the
  corresponding random variables R and ? under this
  transformation?

7
Joint Distribution Functions Example

• The Jacobean is x / ?(x2y2) y / ?(x2y2)
  -y / (x2y2) x / (x2y2) 1 /
  ?(x2y2) 1 / r
• The assumption of equal means gives the joint
  density function of X and Y asf(x,y) 1/2?
  e-(x2y2)/2
• So the joint density function of R and ? isf(r,
  ?) 1/2? re-r2/2 0 lt ? lt 2?, 0 lt r lt ?
• This is known as the Rayleigh Distribution
• Note particularly the independence of ? (I.e.
  uniformly distributed)?

8
Summary

• Pairwise Joint Distribution Functions
• Conditions for a continuous pairwise joint
  distribution to exist
• Formula for the joint probability density
  function
• Example the Rayleigh Distribution
• Transformation of (equal) normal distributions
  from Cartesian to Polar coordinates
• The distribution of ? is uniform and independent
  of R

9
?????