Math 3338
Here are some of the topics discussed so far:
General Theory
•probability (three properties)
•conditional probability: P(A|B) = P(A∩B)/P (B) for P(B)6= 0
•inclusion-exclusion principle: P(A∪B) = P(A) + P(B)−P(A∩B), etc.
•multiplication rule: P(A∩B) = P(B)P(A|B)
•independent events
•Bayes’ theorem
•random variables
–one random variable:
probability mass/density function (p.m.f./p.d.f.)
(cumulative) distribution
histogram, bar graph
–two or more random variables:
joint p.m.f./p.d.f.
marginals
independence
correlation coefficient
distribution of a conditional RV, Y|x
•expected value:
E(u(X)) = Px∈Su(x)f(x), respectively E(u(X)) = R∞
−∞ u(x)f(x)dx
•mean, variance, standard deviation for the population, and a sample
•Moment generating function (m.g.f.): MX(t) = E(etX )
Its connection with the p.m.f./p.d.f., and the moments of X.
•methods of enumeration (§2.2)
•mean, variance and m.g.f. for sums of independent RV’s (in particular:
sums of normal, chi-square, Poisson, etc.)
•the Central Limit Theorem, approximations using CLT
Distributions
For each of the distributions, there is a formula for the p.m.f./p.d.f., and
maybe also for the mean, variance, or m.g.f. There are also natural connections
between these RV’s.
•Discrete:
–uniform
–hypergeometric
–binomial (& Bernoulli trials)
–negative binomial (and geometric)
–Poisson
•Continuous:
–uniform
–Gamma (and exponential, chi-square)
–normal (and relation of chi-square to normal)