stochastic processes chapter 1 definitions UPDATED ACTUAL Questions and CORRECT Answers

stochastic processes chapter 1 definitions UPDATED ACTUAL Questions and
CORRECT Answers


stochastic process a stochastic process is a collection of random variables, {X_t, t ∈ I}. The set I
is the index set of the process. the random variables are defined on a
common state space S.


law of total probability Let B₁, . . ., Bk be a sequence of events that partition the sample space. that
is, the B_i are mutually exclusive (disjoint) and their union is equal to Ω.
then, for any event A,




Bayes' Rule For events A and B,




Conditional Expectation of Y given X = x




Properties of Conditional Expectation 1. (Linearity) For constants a and b and random variables X, Y, and Z,




Properties of Conditional Expectation 2. If g is a function,




Properties of Conditional Expectation 3. (Independence) If X and Y are independent,
E(Y|X = x) = E(Y)


Properties of Conditional Expectation If Y = g(X) is a function of X,
E(Y|X = x) = g(x)