Conditional Probability &
Conditional Expectation
Conditional distributions
Computing expectations by conditioning
Computing probabilities by conditioning
Chapter 3
1
Discrete conditional distributions
Given a joint probability mass function
p  x, y   P  X  x, Y  y
the conditional pmf of X given that Y = y is
p  x, y 
p X Y  x y   P  X  x Y  y 
if pY  y   0
pY  y 
The conditional expectation of X given Y = y is
E  X Y  y    xp X Y  x y 
x
Chapter 3
2
Continuous conditional distributions
Given a joint probability density function f  x, y 
the conditional pdf of X given that Y = y is
f  x, y 
fX Y  x y 
if fY  y   0
fY  y 
This may seem nonsensical since P{Y = y} = 0 if Y is continuous.
Interpret f X Y  x y  dx as the conditional probability that X is between
x and x + dx given that Y is between y and y + dy.
The conditional expectation of X given Y = y is
E  X Y  y    x f X Y  x y  dx

Chapter 3
3
Computing Expectations by Conditioning
Suppose we want to know E[X] but the distribution of X is
difficult to find. However, knowing Y gives us some useful
information about X – in particular, we know E[X|Y=y].
1. E[X|Y=y] is a number but E[X|Y] is a random variable
since Y is a random variable.
2. We can find E[X] fromE  X   EY  E  X Y  
If Y is discrete then E  X    E  X Y  y  pY  y 
y

If Y is continuous then E  X    E  X Y  y  fY  y  dy

Chapter 3
4
Computing Probabilities by Conditioning
Suppose we want to know the probability of some event,
E (this event could describe a set of values for a random
variable). Knowing Y gives us some useful information about
whether or not E occurred.
1 if E occurs
Define an indicator random variable X  
 0 otherwise
Then P(E) = E[X], P(E|Y = y) = E[X|Y = y]
So we can find P(E) from
P  E    P  E Y  y  pY  y 
y
or



P  E Y  y  fY  y  dy
Chapter 3
5
Strategies for Solving Problems
• What piece of information would help you find the
probability or expected value you seek?
• When dealing with a sequence of choices, trials, etc.,
condition on the outcome of the first one
Can also find variance by conditioning:

Var  X   E  Var  X Y   Var E  X Y 
Chapter 3

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