6 Flashcards
N([t1 t2]) = ?
N(t_2) - N(t_1)
Function that counts the # of random events during a specific time?
N(t_x)
Axioms of Poisson Process: N is a poisson process iff? 4•
•N(0) = 0
•Independence: All events within a time interval are independent of those in another time interval.
•Homogeneity: Probabilities do not change if the same length of time interval occurs later.
•Non-concurrence: P{N(h) >= 2} «_space;P{N(h) = 1} when h is small -> P{N(h) = 1} = lambda*h + o(h).
E[N(h)] = ?
lambda*h
If f(t) is o(t), then?
lim(t-> 0)(f(t) / t) = 0 in which it is slower than a linear function.
P{N(t) = 0} = ?
e^(-lambda*t)
Exponential random variable with rate (lambda)?
P{T_1 >= t} = e^(-lambda t), T_1 is the time of the first event
P{N(t) = k} = ?
(e^-(lambda)t (lambda t)^k) / k!
Geometric random variable with parameter p: P(X = k) = ?
q^(k-1) p
!!!Review diff between Poisson, b distribution, and geometric random variables
When to use geometric random variable?
Whenever the question asks how many times until desired event in non-continuous time.
!!!Review continuous times
Expected value of geometric random variable: E[X] = ? 2•
•Sum(k=1)(oo) (q^(k-1) pk), •1/p
E[X - 1] = ? In geometric random variable
q sum(j=0 -> oo) (q^(j - 1) pj = q E[x]
Second momentum of geometric random variable: E[X^2] = ? (2•)
•Sum(k=1 -> oo) (q^(k-1) pk^2), •(2 - p)/p^2
Variance of geometric random variable with parameter p: E[X^2] - (E[X])^2 = ? 2•
(1 - p)/p^2, q/p^2
Negative binomial random variables?
The time it takes for a number of trials of desired event.
Probability mass function of negative binomial random variable: P{X = k} = ?
(k - 1) chose (r - 1) p^(r-1) (1 - p)^(k-r) p, with parameters (p, r) where r is the number of trials that got the desired event.
Expected value of negative binomial random variable: E[X] = ?
r/p
Variance of negative binomial random variable: Var[X] = ?
rq/p^2
Poisson approximation: lambda = ?
p n