Joint and marginal distribution functions

1
Joint and marginal distribution functions

• For any two random variables X and Y defined on
  the same sample space, the joint c.d.f. is
• For an example, see next slide.
• The marginal distributions can be obtained from
  the joint distributions as follows
• When X and Y are both discrete, the joint
  probability mass function is given by
  The probability
  mass function of X, pX(x), is obtained by
  summing over y. Similarly for pY(y).

2
C.D.F. for a Bivariate Normal (density shown
later)
3
Example for joint probability mass function

• Consider the following table
• Using the table, we have

Y0 Y3 Y4

X5 1/7 1/7
1/7 3/7
pX
X8 3/7 0
1/7 4/7
4/7 1/7 2/7

pY
4
Expected Values for Jointly Distributed Random
Variables

• Let X and Y be discrete random variables with
  joint probability mass function p(x, y). Let the
  sets of values of X and Y be A and B, resp. We
  define E(X) and E(Y) as
• Example. For the random variables X and Y from
  the previous slide,

5
Law of the Unconscious Statistician Revisited

• Theorem. Let p(x, y) be the joint probability
  mass function of discrete random variables X and
  Y. Let A and B be the set of possible values of
  X and Y, resp. If h is a function of two
  variables from R2 to R, then h(X, Y) is a
  discrete random variable with expected value
  given by
  provided that the sum is absolutely convergent.
• Corollary. For discrete random variables X and
  Y,
• Problem. Verify the corollary for X and Y from
  two slides previous.

6
Joint and marginal distribution functions for
continuous r.v.s

• Random variables X and Y are jointly continuous
  if there exists a nonnegative function f(x, y)
  such that for every
  well-behaved subset C of lR2. The function f(x,
  y) is called the joint probability density
  function of X and Y.
• It follows that
• Also,

7
Density for a Bivariate Normal (see page 449 for
formula)
8
Example of joint density for continuous r.v.s

• Let the joint density of X and Y be
• Prove that (1)
  PXgt1,Ylt1 e1(1 e2) (2)
  PXltY 1/3 (3) FX(a) 1
  ea, a gt 0, and 0 otherwise.

9
Expected Values for Jointly Distributed
Continuous R.V.s

• Let X and Y be continuous random variables with
  joint probability density function f(x, y). We
  define E(X) and E(Y) as
• Example. For the random variables X and Y from
  the previous slide, That
  is, X and Y are exponential random variables.
  It follows that

10
Law of the Unconscious Statistician Again

• Theorem. Let f(x, y) be the joint density
  function of random variables X and Y. If h is a
  function of two variables from lR2 to lR, then
  h(X, Y) is a random variable with expected value
  given by
  provided the integral is absolutely convergent.
• Corollary. For random variables X and Y as in
  the above theorem,
• Example. For X and Y defined two slides previous,

11
Random Selection of a Point from a Planar Region

• Let S be a subset of the plane with area A(S). A
  point is said to be randomly selected from S if
  for any subset R of S with area A(R), the
  probability that R contains the point is
  A(R)/A(S).
• Problem. Two people arrive at a restaurant at
  random times from 1130am to 1200 noon. What is
  the probability that their arrival times differ
  by ten minutes or less? Solution. Let X and Y
  be the minutes past 1130 am that the two people
  arrive. Let
  The desired probability is

12
Independent random variables

• Random variables X and Y are independent if for
  any two sets of real numbers A and B,
  That is,
  events EA X? A, EBY? B are independent.
• In terms of F, X and Y are independent if and
  only if
• When X and Y are discrete, they are independent
  if and only if
• In the jointly continuous case, X and Y are
  independent if and only if

13
Example for independent jointly distributed r.v.s

• A man and a woman decide to meet at a certain
  location. If each person independently arrives
  at a time uniformly distributed between 12 noon
  and 1 pm, find the probability that the first to
  arrive has to wait longer than 10 minutes.
  Solution. Let X and Y denote,
  resp., the time that the man and woman arrive. X
  and Y are independent.

14
Sums of independent random variables

• Suppose that X and Y are independent continuous
  random variables having probability density
  functions fX and fY. Then
• We obtain the density of the sum by
  differentiating
• 
  The right-hand-side of the latter
  equation defines the convolution of fX and
  fY.

15
Example for sum of two independent random
variables

• Suppose X and Y are independent random variables,
  both uniformly distributed on (0,1). The density
  of XY is computed as follows
• Because of the shape of its density function, XY
  is said to have a triangular distribution.

16
Functions of Independent Random Variables

• Theorem. Let X and Y be independent random
  variables and let g and h be real valued
  functions of a single real variable. Then
  (i) g(X) and h(Y) are also independent
  random variables
• Example. If X and Y are independent, then