MAT 3301681 Lecture 13 3/17/2021

Happy

foRandlR)St

.Patrick 's Day !

A random variable X :L → IR is a aartinuousrardo-ma.dk

if there exists a nonnegative function to

f :D - [o.to?~mustsgtfistz:ffjxidx-a .

called probability density function such that :'

-I n f

PCXEB) -- fpgflxldx i.MY,where Bcr is any fneasvroble)

subset.

Note: p(X=a)=J!fHdx=0planning:*. on:÷i¥÷

- n Eas b Eta 70

temporal : Pl - next a) =LII

ft! flxldx

EI: Suppose X : IR→ (Oita)is a random variable with prob . density

function flx) - Max 10 , CLI- 4x4}; where C is some real number.

a) find the value of CHR so that fix) is a p - d. f .

b) Compute PIX> o) and PCX> 44 ) a

•

C

a) Need -441¥ ,

to - Yz; i

,

I --ffk7dx=f.dxi-%fddxtf.tn?qdx--f!yIHdx-a=O--cf!Ik-4x)dx=2cfx-4×37112=241 -¥¥)

= 2ft- E) c - Esc ⇒ c=±7"

plxzoj-fodfkddx-fok.EU -4x4 dx = - - - =II

XELO ,ta)

Plx=S!µ%adx .fm?zk-4x4dx--Zfx-ax5Hiii=-- .I

X -444 ,t§. .

- =I@MA

Cumulative Distribution Function (c.D.F.)-

X cont . random variable,f :D → [Oita) prob . density function

x( p -

d. f.)

Ffx) :=P(XEx)=f ftldtw

I- a

XEL- ax]f

T-und.thm.caleuhs.FI/x)=flx) . I i

i 'llHunn

a- { a'at EzCf . atef Tylor

Pla - EEXEATE) - f find Ef④eIJije."

- a-42← "

density "

prob . of"

finding"

X ddzfa"

"Icadx={Hate) .

in a small region qq.ec, g.again.# a.⇒E:# kit:i" "Ii:# ""

of size E>o near a

Exe. Suppose the lifetime (in hours) of a certain circuit

in an electronic device is a random variable w/ p .d. f .

A -

i 43fix' - I :o÷ ! iii:

'

roof - - - yo ,suppose the device has 5 circuits

. too

what is the prob .

that exactly 2 of these circuits

will need to be replaced within the first 150 hours

of operation? (Assume the circuits fail independently )

X = lifetime of a circuit . ( cont . random . variable)150 150

P(Xo)= J flxldx - f TEDx - roof-⇒ It"

I - p100 too

Xe -4,1503←

prob. that I

Mycircuit fails

= Ioof - + Foo) = too (-2*3)--12 in the first150 hours

.

Yn Binomial ( 5,13)PH -2) - (E) ¥5 = .

.- =E€f,-

Recall : if X is discrete rand . var.

E- (x) , ⇐ xp , Efg =§gkdpkd , Ver - ECX') - ENT

For a cart . random variable X with pelf fix) :to

E- (X) - J x - fkddx , Efg(XD .- ftgglxflxldx- N

(as before : Vara) - Efx') - Efx)?)

Revisit the 1st example : ft) - max 30 , Elt - 4x4 )A

•

C

EWS!ix¥d"fix) s

e- ZS!:X -4x3dx=ZfE- x4)=o .

"e"

ENT -- * I÷¥⇒d×=3f!ZxHx4dx=ff¥¥E) )!= I - ¥¥g - Ift - ZHI - E

← am:%I+

varlxt-EHY-E.IE -IIF Ox - faiths '

Uniform distribution-

A continuous random variable X ismmformlydistribded-ifitsp-d.fr. only assumes 2 valves : O and c .

A

tooEH) fix)

1=1 fix)dx=(b-a) - c c -F-

-

.all I

.

# i-0 b-

⇒ c = Ia a- b

b- a

f = {0 if k$643 "

all points are

÷ if Xe Laib] . equally likely"

Elms! ÷dx- Ea .EE - Ea . '=a'

9

Var (x) - Efx') - Efxjaverage of

asb.

HM - six's'÷;dx=s÷'El! }%÷=b:;

Varlxtbiabzta - fatty! la7a}tblahatf-7.3-tzkkkaabf-ca.LI?=tEaaIbnIik'¥1.