Download Section 3.1

Random Variables
• If  is an outcome space with a probability measure
and X is a real-valued function defined over the
elements of , then X is a random variable.
• Standard notation
– Capital letter for a random variable (e.g., X)
– Lower-case letter for a realization of the random variable
(e.g., x)
Example
• Flip a coin until the first tail or until the 4th
flip, whichever comes first. Let X represent
the number of heads observed.
– What’s the range of X?
– What’s the probability distribution of X?
Joint Distributions
• Given two random variables X and Y defined
for in the same setting, we can consider the
joint outcome (X, Y) as a random pair of
values.
– The event (X = x, Y = y) is the intersection of the
events (X = x) and (Y = y).
– The distribution of (X, Y) is called the joint
distribution of X and Y.
Marginal Probabilities &
Distributions
• Given a joint distribution of X & Y:
– The marginal probability that X = x is
P  X  x    P  x, y 
all y
– The distribution of X (irrespective of Y) is called
the marginal distribution of X.
• As x varies over the range of X, the marginal
probabilities that X = x define the marginal distribution
of X.
Conditional Distributions
• Given X = x, as y varies over the range of Y
the probabilities P(Y=y|X=x) define a
probability distribution over the range of Y.
• This distribution is called the conditional
distribution of Y given X = x.