(a) Let {N t > O) be a nonhomogeneous Poisson measure with mean worth capacity

m(t). Given = n, show that the unordered arrangement of appearance times has a similar dissemination as n

free and indistinguishably appropriated irregular factors having circulation work

m(x)

1,

(b) Suppose that workers bring about mishaps in understanding vvith a nonhomogeneous Poisson

measure with mean worth capacity m(t). Assume further that each harmed man is out ofwork peak a

irregular sum oftime having dispersion F. Let X(t) be the number ofworkers vvho are out of

work at time t. ay utilizing part find

Q42

Let Xl, X2, be free and indistinguishably dispersed nonnegative nonstop arbitrary

factors having thickness work f(x). We say that a record happens at time n If Xn is bigger than

each ottne past qualities x l, . .. , X . (A record consequently happens at time 1.) If a

record happens at time n, at that point Xn is known as a record esteem. As such, a record happens

at whatever point another high is reached: and that new high is known as the record esteem. Let N(t) indicate

the quantity of record esteems that are not exactly or equivalent to t. Describe the interaction

(N (t), t 2 0) when (a) fis a self-assertive ceaseless thickness work.

(a) f is a self-assertive ceaseless thickness work.

(b) f (x) Xei-r

Clue: Finish the accompanying sentence: There will be a record whose worth is among t and t + dtif

the principal Xi that is more noteworthy than t lies between

Q43

In great years, storms happen as per a Poisson cycle with rate 3 for every unit time, while in

different years they happen as per a Poisson cycle with rate 5 for every unit time. Assume straightaway

year will be a decent year witn likelihood 0.3. Let Od(t) indicate the quantity of tempests during the first

t time units of one year from now.

(a) Find

(b) Is {N(t)} a Poisson interaction?

(c) Does {N(t)) have fixed augmentations? VVhy or why not?

(d) Does it have free augmentations? Why or why not?

(e) It one year from now begins otfvvith three tempests by time t = 1, what is the restrictive likelihood it is a

great year?

Q44

A few segments of a two-segment framework flop in the wake of getting a stun. Stuns ot three sorts

show up autonomously and as per Poisson measures. Stuns ot the main sort show up at

a Poisson rate and cause the principal segment to come up short. Those of the subsequent kind show up at a

Poisson rate 2 and cause the subsequent segment to fall flat. The third sort of stun shows up at a

Poisson rate 3 and causes botn segments to fall flat. Let X, and X2 indicate the endurance times for

the two parts. Show that the joint circulation of Xl and X2 is given by

PIXI > s. Xl > = i3 max(s.t)}

This appropriation is knovvn as the bivariate exponentja/conveyance.