Lecture 14: Hazard - Duke University

Lecture 14: Hazard

Statistics 104

Colin Rundel

March 12, 2012

Gamma Function

Hazard Review

We have just shown the following that when X Exp():

E (X n )

=

n! n

Lets set = 1 and define an new value = n + 1

E(X-1) = ( - 1)!

x-1e-xdx = ( - 1)!

0

() x-1e-xdx = ( - 1)!

0

Using a tradition definition of the factorial it only makes sense when n N

but we can use this new definition of the gamma function () for any

R+

Statistics 104 (Colin Rundel)

Lecture 14

March 12, 2012 1 / 25

Hazard Review

Gamma/Erlang Distribution - CDF

Imagine instead of finding the time until an event occurs we instead want to find the distribution for the time until the nth event.

Let Tn denote the time at which the nth event occurs, then Tn = X1 + ? ? ? + Xn where X1, . . . , Xn iid Exp(). Let N (t) be the number of events that have occured at time t.

F (t) = P (Tn t) = P (N (t) n)

= P (N (t) = j)

j=n

e-t(t)j =

j!

j=n

Statistics 104 (Colin Rundel)

Lecture 14

March 12, 2012 2 / 25

Hazard Review

Gamma/Erlang Distribution - pdf

d

d e-t(t)j

f (t) = F (t) =

dt

dt

j!

j=n

e-t(t)j-1

e-t(t)j

=

j

-

j!

j!

j=n

e-t(t)j-1 e-t(t)j

=

-

(j - 1)!

j!

j=n

j=n

e-t(t)j-1

e-t(t)j-1 e-t(t)j

=

+

-

(j - 1)!

(j - 1)!

j!

j=n+1

j=n

e-t(t)n-1 e-t(t)j e-t(t)j

=

+

-

(n - 1)!

j!

j!

j=n

j=n

e-tntn-1 e-tntn-1

=

=

(n - 1)!

(n)

Statistics 104 (Colin Rundel)

Lecture 14

March 12, 2012 3 / 25

Erlang Distribution

Hazard Review

Let X reflect the time until the nth event occurs when the events occur according to a Poisson process with rate , X Er(n, )

e-x n xn-1 f (x|n, ) =

(n - 1)! e-x(x)j F (x|n, ) =

j!

j=n

n MX (t) = - t

E(X) = n/ V ar(X) = n/2

Statistics 104 (Colin Rundel)

Lecture 14

March 12, 2012 4 / 25

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