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