Preliminaries Flashcards
(19 cards)
What is a stochastic process?
A stochastic process is a process which evolves randomly over time (or space, or both).
What does ‘stochastic’ mean?
Stochastic is equivalent to ‘random’.
What does ‘process’ mean in a stochastic process?
Process refers to something that occurs over time (or space, or both).
What is the main focus of the course on stochastic processes?
The course focuses on developing mathematical tools to study the properties of stochastic processes and their long-term behavior.
Give an example of an application of stochastic processes.
Applications of stochastic processes are wide-ranging, e.g., in physics, engineering, social science, genetics, and finance.
What is a discrete-time process?
A discrete-time process is {X_n, n ∈ N}, where N = {0, 1, 2, …}, with time represented by natural numbers.
What is a continuous-time process?
A continuous-time process is {X(t), t ∈ T}, where T = R+ (i.e. t ≥ 0), and time can take any value in the positive reals.
What is an example of a discrete-time, discrete state-space process?
An example is tossing a coin and counting the number of excess heads (how many more heads than tails).
What is a random variable?
A random variable is a variable whose numerical values are randomly chosen from a set of possible values governed by a probability distribution.
What is the sample space in a coin-tossing experiment?
The sample space for tossing a coin twice is Ω = {HH, HT, TH, TT}.
What does the probability model assign to each sample point?
It assigns a probability to each sample point, denoted p(ω), where ∑(p(ω)) = 1.
What is the distribution function of a random variable?
The distribution function of a random variable X is F_X(x) = P(X ≤ x).
What is a discrete random variable?
A discrete random variable takes only finitely many or countably infinitely many values.
What is the variance of a discrete random variable X?
The variance of X is Var(X) = E[(X − E(X))^2] = E(X^2) − [E(X)]^2.
What is the probability mass function (PMF) of a discrete random variable?
The PMF of X is P(X = x) = P({ω : X(ω) = x}).
What is the expectation of a continuous random variable X?
The expectation of X is E(X) = ∫∞−∞ x f_X(x) dx.
What is the moment generating function (MGF) of a random variable?
The moment generating function of X is M_X(s) = E[exp(tX)].
What is the use of a probability generating function (PGF)?
A PGF is used to wrap up the sequence of probabilities of a discrete random variable into a single expression for easier calculation of moments and probabilities.
What is the PGF of a geometric distribution?
The PGF of a geometric distribution with parameter p is G_X(s) = p s / [1 - (1 - p) s].
