Professor Levine
Discrete vs. Continuous Distributions
In the following table, S denotes a discrete state space, IR denotes the real line, g(X) is a real-valued
function of the random variable X, and H(X, Y ) is a real-valued function of the random variables X and
Y.
Discrete, x ∈ S
Continuous, x ∈ IR
E{g(X)}
pmf f (x) = P (X = x)
P
x∈S P (X = x)
Pb
x=a P (X = x)
P
x∈S xP (X = x)
P
x∈S g(x)P (X = x)
pdf f (x), P (X = x) = 0
Rx
f (x) dx
−∞
Rb
f (x) dx
a
R
IR xf (x) dx
R
IR g(x)f (x) dx
Bivariate
Discrete, x, y ∈ S
Continuous, x, y ∈ IR
Rb Ra
f (x, y) dxdy
−∞ −∞
R R
IR IR h(x, y)f (x, y) dxdy
R
IR f (x, y) dxdy
Univariate
pmf/pdf
cdf F (x) = P (X ≤ x)
P (a ≤ X ≤ b)
E(X)
P
P (x ≤ a, Y ≤ b)
E{h(X, Y )}
marginal, f (x)
conditional, f (x|y)
P
x∈S
x≤a
P
y≤b
P
y∈S
P
y∈S
P (X = x, Y = y)
h(x, y)P (X = x, Y = y)
P (X = x, Y = y)
P (X = x, Y = y)/P (Y = y)
f (x, y)/f (y)
independence
P (X = x, Y = y) = P (X = x)P (Y = y)
f (x.y) = f (x)f (y)
E(X|Y = y)
E{g(X)h(Y )} = E{g(X)}E{h(Y )}
P
x∈S xP (X = x|Y = y)
E{g(X)h(Y )} = E{g(X)}E{h(Y )}
R
IR xf (x|y) dx