Math 3280 20 - lo - 19

Review.

• Normal r - v X with parameters fu and 62 ;

fuk ÷, e-"""%"

, xefo.es ,

X -µZi=-g is a standard normal nu

.

( with mean 0,variance 't )

• DeMoipaaThn : Given pelo, 't) ,let Xn be a binomial rv with parameters (n, p) .

ThenX. n - up

Pf as

v¥p,s b }

→ Sabata ax

as n→ is.

• Exponential v.v with parameter d ( mo ),

fix , = f de""

, if x > o

o, otherwise

,

Suppose that the length of a phone call in minutes is an exponential random variable with parameter λ = 1/10 . If someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait(a) more than 10 minutes;(b) between 10 and 20 minutes.

Example I .

Solution :

Let X denote the waiting time of theperson ( in minutes) .

According to our assumption, X is an exponentialrv with parameter D= To .

Hence

(as p { X s. to }= f ! fan dx = f

,! de-"dx

= - e-''Y's

10

= e-""

= e-it

.

Cb ) P { 10 EX Ezo }

= J! de-" dx=

- e-" I ! = e-" "

- e

""

= e- ' - e-2

. he

§ 5.7 The distribution of a function of acts r. u .

Q : Let X be a cts r. u .

with density f×.

Let Gi IR→ R be a function .Let Y - GH) .

How to find the distribution of Y ?

Exerz . Let X be a cts r. u.with density fx

.

Ket Y= X?

Find the pdf of Y.

Solution : We first calculate the CDF of Y :

Eyes ) = Pf Ye y }

=p { X 's y )

= / o if y so

( Pf -rye Xeafy } if yzo

= Extra) - Fxfry)

Taking derivative of Fy with respect to Y gives

o if y so

fyl" = {fxagy.gg#xtrd:'try it Y > o

Ex 3.

Let X be an exponential r-u withparameter d .

Let 4=4Find the Pdf of Y .

Solution : first notice that

PI Keo } = 0

Now

Fyn =p { Ye y }= pg # ⇐ y }

=gP{ Katy } if y > o

o if y so

= /t - Exc 'T ) if y > o

o otherwise

Taking derivative of Fy wrt y gives

fycsi-f-fxttl.tt) y > o

o otherwise

= fifty if y > o

{o otherwise

=

gd. e-"T.fr if y > o

o otherwise,

prop.tt : let X be acts v.v. with

density f-× .

Let g : IR→ IRbe a differentiable , strictly monotone

function .

Let Y - G ( X ),

then

the Pdf of Y is given by

fycsi-fxfg.tn/./dd5yI/{ if 5- gag !: ×

0 otherwise.

Pf . Suppose that g is strictly increasing.

Then

Fycyt = PE Yey }= pg gypsy }

=/'t it Y . 7.Einar"

P↳¥yerayeH)o if ysmingcxi

KEIR

P{ He g- 'on }= Ex ( gin) .

Taking derivative of Fy gives

fyCy)={0 it Yet Raycg ,

f-* faith . ) . daft ,

if y = gunfor some X .

me .