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Probability

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Product Rule:

• P(A|B) = P(AB)/P(B)
• P(AB) = P(A|B) P(B)

Bayes Rule:

• Conditional Probability

Ch4 - Random Variables

A Random Variable - X

• Where X is a set of {x1, x2,.. xn}
• We say that X maps S into Lambda

Probability Mass Function - p(x)

• For a discrete random variable, the function p(x) denotes the probability of an event x (e.g. the sum of dice = 5).
• Note that p(x) denotes the PMF of an element x in X, whereas P(E) or Pr(E) denote the probability of an event E in S.
• All values in p(x) must sum to one.
• If i is not an element of the PMF, then its probability is 0.

• Example: The pmf of the sum of two dice.

Cumulative Distribution Function = F(x)

• For each i, the CDF is the sum of p(n) such that 0 <= n <= i

Expected Value of X = E[X]

• The weighted sum of all PMF values
• E[X] = Sum p(x)*x for all x in X
• Example: if X = {1, 2, 3} and p(x) = {.25, .50, .25}, then E[X] = 1*(.25) + 2*(.50) + 3*(.25)

Variance = Var(x) = E[ (x - E[x])^2 ]

• We can also write this out in terms of sums within Lambda-sub-g or Lambda.
• Variance is also called second moment, where E[ (x - E[x])^k ] = the kth moment
• We square the variance in order to allow it to equally weight numbers on either side of the expected value.
• Variance of a fair die is 35/12, the expected value is 7/2