Conditionality and stopping times in probability
Mark Osegard, Ben Speidel, Megan Silberhorn, and Dickens Nyabuti
Conditional Expectation
Conditional Probability
fX ( y)
Conditional Expectation
Discrete: E X | Y y xPX x | Y y
x
Continuous: E X | Y y xfX | Y(x, y)dx
Note:
y E X | Y y is a function
of y. We write this as E X | Y
Conditional Variance
Definition
Var(X | Y ) E X EX | Y 2 | Y
E X 2 | Y EX | Y 2
VarX EVarX | Y VarEX | Y
Proof
using
EX EEX | Y
since
VarX | Y E X 2 | Y EX | Y 2
X n1, X n2 ,...
Independence
Summarizing the idea of stopping times with Random Variables we see that the decision made to stop the sequence at Random Variable N depends solely on the values of the sequence
E EX | Y 2 EX 2
…adding
EVarX | Y VarEX | Y EX 2 EX 2
VarX
Thus we' ve shown that
VarX EVarX | Y VarEX | Y g
Stopping times
Stopping Times
Definition Application to Probability Applications of Stopping Times to other
EX
|Y
y fY y dy
xfX
|
Yx
|
ydx
fY y dy
xfX | Yx | y fYydxdy
xБайду номын сангаас
f
x, y
fY y
fY y dxdy
Continuous Case Cont.
xf x, ydxdy xf x, ydydx
(Fubini’s Theorem)
taking expectations of both sides
EVarX | Y E EX 2 | Y EX | Y 2 EEX 2 | Y E EX | Y 2 EX 2 E EX | Y 2
Note as well …
VarEX | Y E EX | Y 2 EEX | Y 2
formulas
Stopping Times
Basic Definition: A Stopping Time for a process does exactly that, it tells the process when to stop. Ex) while ( x != 4 ) {…} The stopping time for this code fragment would be the instance where x does equal 4.
Stopping Times: A Discrete Case
From our previous slide we have the sequence:
X1, X 2 , X 3,...
A discrete Random Variable N is a stopping time for this sequence if : {N=n} Where n is independent of all following items in the sequence
Proof: Continuous Case
Recall, if X,Y are jointly continuous with
joint pdf f x, y
Define:
fX | YX | Y
f x, y fY y
and EX | Y y xfX | YX | Y dx
Note:
x f x, ydydx f x, ydy
So,
xfXxdx EX
Therefore, concluding
EX EEX | Y
Summary:
When Y is discrete,
EX | Y yPY y
y
When Y is continuous,
EX | Y yfYydy
i.e. EX | Y y EX | Y y
(Conditional Expectation Function)
Theorem:
EX EEX | Y
Clearly, when Y is discrete,
EX | Y yPY y
y
When Y is continuous,
EX | Y yfYydy
Stopping times in Sequences
Define: Suppose we have a sequence of Random Variables (all independent of each other) Our sequence then would be:
X1, X 2 , X 3,...
Discrete: Conditional Probability Mass Function
PX x | Y y PX x,Y y PY y
Continuous: Conditional Probability Density Function
f (x, y) fX | Y(x | y) :