6

Question 1

N([t1 t2]) = ?

Answer 1

N(t_2) - N(t_1)

Question 2

Function that counts the # of random events during a specific time?

Answer 2

N(t_x)

Question 3

Axioms of Poisson Process: N is a poisson process iff? 4•

Answer 3

•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} &laquo_space;P{N(h) = 1} when h is small -> P{N(h) = 1} = lambda*h + o(h).

Question 4

E[N(h)] = ?

Answer 4

lambda*h

Question 5

If f(t) is o(t), then?

Answer 5

lim(t-> 0)(f(t) / t) = 0 in which it is slower than a linear function.

Question 6

P{N(t) = 0} = ?

Answer 6

e^(-lambda*t)

Question 7

Exponential random variable with rate (lambda)?

Answer 7

P{T_1 >= t} = e^(-lambda t), T_1 is the time of the first event

Question 8

P{N(t) = k} = ?

Answer 8

(e^-(lambda)t (lambda t)^k) / k!

Question 9

Geometric random variable with parameter p: P(X = k) = ?

Answer 9

q^(k-1) p

Question 10

!!!Review diff between Poisson, b distribution, and geometric random variables

Question 11

When to use geometric random variable?

Answer 11

Whenever the question asks how many times until desired event in non-continuous time.

Question 12

!!!Review continuous times

Question 13

Expected value of geometric random variable: E[X] = ? 2•

Answer 13

•Sum(k=1)(oo) (q^(k-1) pk), •1/p

Question 14

E[X - 1] = ? In geometric random variable

Answer 14

q sum(j=0 -> oo) (q^(j - 1) pj = q E[x]

Question 15

Second momentum of geometric random variable: E[X^2] = ? (2•)

Answer 15

•Sum(k=1 -> oo) (q^(k-1) pk^2), •(2 - p)/p^2

Question 16

Variance of geometric random variable with parameter p: E[X^2] - (E[X])^2 = ? 2•

Answer 16

(1 - p)/p^2, q/p^2

Question 17

Negative binomial random variables?

Answer 17

The time it takes for a number of trials of desired event.

Question 18

Probability mass function of negative binomial random variable: P{X = k} = ?

Answer 18

(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.

Question 19

Expected value of negative binomial random variable: E[X] = ?

Answer 19

r/p

Question 20

Variance of negative binomial random variable: Var[X] = ?

Answer 20

rq/p^2

Question 21

Poisson approximation: lambda = ?

Answer 21

p n