Module 2

Question 1

Survival function

Answer 1

S(x) = 1 - F(x)

Question 2

Hazard Rate

Answer 2

u(x) = -S’(x)/S(x)

Question 3

sum of exponentials with same parameter, is distributed as

Answer 3

Gamma(n,λ)

Question 4

Min of a set of exponentials is distributed as

Answer 4

Exp ( ∑ λ)

λ is the parameter of each exponential

Question 5

wrt Poison, exponential dists model…

Answer 5

the time between each event

Question 6

E(x) ( x is exponetial)

Answer 6

=1/λ

Question 7

Indepentand increments expression in counting process

Answer 7

N(t) - N(s) is independant of N(s)

Question 8

Tn in a poison represents

Answer 8

Time elapsed between event n-1 to nth

Question 9

Sn, waiting time to nth event (Poisson process) can be modelled by,

Answer 9

Gamma(n,λ)

Question 10

Poisson Process conditions

Answer 10

N(0)=0
Independant Increments
Stationary Increments

Question 11

Poisson process Stationary Increments meaning

Answer 11

The # of events in an interval is only dependant on its length.

Question 12

Sum of Poisson processes means

Answer 12

N1 and N2 are 2 independant processes. N=N1+N2 is dist Poisson(λ1+λ2)

Question 13

Thinning/Split Poisson means

Answer 13

If for each event, it will fall into a category with prob p,

Then events in that category will occur with Poisson(p*λ)

Question 14

Different feature about Non-homogenious Poisson Process’

Answer 14

They have unit ‘unit jumps’

Question 15

Different type of Poisson processes to be aware of;

Answer 15

1. Sum of Pp
2. thinning of Pp
3. Non-homogenious Pp
4. Compound Pp

Question 16

Good analogy to explain Compound Pp

Answer 16

Filling up a damn. Timing is N(t), (amount of rain each time is distributed differently though).

Question 17

Variance of Exponential

Answer 17

1 / λ^2

Question 18

Sum of iiid exponentials is distributed as…

Answer 18

Convolution of exponentials distribution

Question 19

function little o(h) means

Answer 19

lim h->0,

f(h)/h = 0

Question 20

Definition 2 of a poison process (part 4)

Answer 20

It satisfies;
P[ N(t+h) - N(t) = 1 ] = λh + o(h)
and
P[ N(t+h) - N(t) ≥ 2 ] = o(h)

Question 21

Sum of exponentials distributed as;

Answer 21

Gamma(n,lambda)