1. Home
  2. Applied probability
  3. Conditional expectation,
  4. Conditional expectation,

Conditional expectation, regression  (Page 4/9)

<< Chapter < Page
  Applied probability     Page 4 / 9
Chapter >> Page >
f = '(6/5)*(t + 2*u).*(u>=t)'; % Density as string variable tuapprEnter matrix [a b] of X-range endpoints [0 1]Enter matrix [c d] of Y-range endpoints [0 1]Enter number of X approximation points 200 Enter number of Y approximation points 200Enter expression for joint density eval(f) % Evaluation of string variable Use array operations on X, Y, PX, PY, t, u, and PEYx = sum(u.*P)./sum(P); % Approximate values eYx = (4 + 3*X - 7*X.^3)./(6*(1 + X - 2*X.^2)); % Theoretical expressionplot(X,EYx,X,eYx) % Plotting details (see [link] )

— □

Figure two is a graph titled, theoretical and approximate conditional expectation. The horizontal axis is labeled, t, and the vertical axis is labeled E[X | Y = t]. The values on the horizontal axis are from 0 to 1 in increments of 0.1. The values on the vertical axis range from 0.65 to 1 in increments of 0.05. There is a caption inside the graph that reads fXY (t, u) = (6/5)*(t + 2u), for 0 ≤ t ≤ u ≤ 1. There are two plots on this graph. The first is a solid, smooth line labeled Approximate. the second is a smooth, dashed line, labeled theoretical. Both lines follow the same path on the graph, and are so closely fitted that they are nearly indistinguishable. They begin on the lower left side, at approximately (0, 0.67), and continue towards the right with a slightly negative slope for a very small segment, until approximately (0.08, 0.66), where the plots begin gradually increasing at an increasing rate. By midway across the graph, at approximately (0.4, 0.74), the slope of the graph remains positive and constant, and continues in a linear fashion from this point to the top-right corner of the graph, at (1, 1). Figure two is a graph titled, theoretical and approximate conditional expectation. The horizontal axis is labeled, t, and the vertical axis is labeled E[X | Y = t]. The values on the horizontal axis are from 0 to 1 in increments of 0.1. The values on the vertical axis range from 0.65 to 1 in increments of 0.05. There is a caption inside the graph that reads fXY (t, u) = (6/5)*(t + 2u), for 0 ≤ t ≤ u ≤ 1. There are two plots on this graph. The first is a solid, smooth line labeled Approximate. the second is a smooth, dashed line, labeled theoretical. Both lines follow the same path on the graph, and are so closely fitted that they are nearly indistinguishable. They begin on the lower left side, at approximately (0, 0.67), and continue towards the right with a slightly negative slope for a very small segment, until approximately (0.08, 0.66), where the plots begin gradually increasing at an increasing rate. By midway across the graph, at approximately (0.4, 0.74), the slope of the graph remains positive and constant, and continues in a linear fashion from this point to the top-right corner of the graph, at (1, 1).
Theoretical and approximate conditional expectation for above .

The agreement of the theoretical and approximate values is quite good enough for practical purposes. It also indicates that the interpretation isreasonable, since the approximation determines the center of mass of the discretized mass which approximates the center of the actual mass in each vertical strip.

Extension to the general case

Most examples for which we make numerical calculations will be one of the types above. Analysis of these cases is built upon the intuitive notion of conditional distributions.However, these cases and this interpretation are rather limited and do not provide the basis for the range of applications—theoretical and practical—which characterizemodern probability theory. We seek a basis for extension (which includes the special cases). In each case examined above, we have the property

( A ) E [ I M ( X ) g ( Y ) ] = E [ I M ( X ) e ( X ) ] where e ( t ) = E [ g ( Y ) | X = t ]

for all t in the range of X .

We have a tie to the simple case of conditioning with respect to an event. If C = { X ∈ M } has positive probability, then using I C = I M ( X ) we have

( B ) E [ I M ( X ) g ( Y ) ] = E [ g ( Y ) | X ∈ M ] P ( X ∈ M )

Two properties of expectation are crucial here:

  1. By the uniqueness property (E5) , since (A) holds for all reasonable (Borel) sets, then e ( X ) is unique a.s. (i.e., except for a set of ω of probability zero).
  2. By the special case of the Radon Nikodym theorem (E19) , the function e ( · ) always exists and is such that random variable e ( X ) is unique a.s.

We make a definition based on these facts.

Definition . The conditional expectation E [ g ( Y ) | X = t ] = e ( t ) is the a.s. unique function defined on the range of X such that

( A ) E [ I M ( X ) g ( Y ) ] = E [ I M ( X ) e ( X ) ] for all Borel sets M

Note that e ( X ) is a random variable and e ( · ) is a function. Expectation E [ g ( Y ) ] is always a constant. The concept is abstract. At this point it has little apparent significance, except that it must include the two special cases studied in the previoussections. Also, it is not clear why the term conditional expectation should be used. The justification rests in certain formal properties which are based on the defining condition(A) and other properties of expectation.

In Appendix F we tabulate a number of key properties of conditional expectation. The condition (A) is called property (CE1) . We examine several of these properties. For a detailed treatment and proofs, any of a number of books on measure-theoretic probability may be consulted.

(CE1) Defining condition . e ( X ) = E [ g ( Y ) | X ] a.s. iff
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask