Continuous Random Variables
(most slides borrowed with permission
from Andrew Moore of CMU and Google)
http://www.cs.cmu.edu/~awm/tutorials
Announcements
• CS Welcome event
– Thursday 3:30, ECCR 265
– poster presentations
• Mozer lab research meeting
– Wednesdays 11:00-12:30, ECCS 127
– Email me if you’d like to be on our mailing list
Real-Valued Random Variables
• Previous lecture on probability focused on
discrete random variables
– true, false
– male, female
– freshman, sophomore, junior, senior
• Can sometimes quantize real variables to make
them discrete
– E.g., age, height, distance
• Today: how to handle variables that cannot be
quantized
Probability Mass Vs. Density
• Discreet RVs have a probability mass
associated with each value of the variable
– P(male)=.7, P(female)=.3
• Imagine if the variable
had an infinite
number of values
instead of a finite
number…
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
0.12
0.06
0.1
0.05
0.08
0.04
0.06
0.03
0.04
0.02
0.02
0.01
0
0
Probability Mass Vs. Density
• Continuous RVs have a probability density
associated with each value
– Probability density function (PDF)
• Density is derivative of mass
• Notation:
P(…) for mass,
p(…) for density
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
0.12
0.06
0.1
0.05
0.08
0.04
0.06
0.03
0.04
0.02
0.02
0.01
0
0
= E[X2] - E[X]2
Density estimate of automobile
weight and MPG
Note change in
notation: Previously
used P(x^y) for
joint
= E[(Xi - mi )(Xj - m j )]
Covariance Facts
Consider 2D case with (X,Y)
s xy2 = E[(Xi - mi )(X j - m j )] = E[Xi X j ] - mi m j
s x2 = E[(Xi - mi )(Xi - mi )] = E[Xi2 ] - mi2
FALSE
TRUE
?
?
Mike’s Basic Advice on Continuous
Random Variables
• Ignore the fact that p(x) is a probability density
function and treat it just as a mass function,
and the algebra all works out.
• Alternatively, turn densities to masses with dx
terms, and they should always cancel out.
• Don’t be freaked when you see a probability
density >> 1.
• Do be freaked if you see a probability mass or
density < 0.