Random variables Some random eperimens may yield a sample space whose elemens evens are numbers, bu some do no or mahemaical purposes, i is desirable o have numbers associaed wih he oucomes A random variable is a funcion ha assigns a real number, ζ, o each oucome ζ in he sample space of a random eperimen The sample space S is he domain of he random variable and he se S of all values aken on by is he range of he random variable Noe ha S R, R is se of all real numbers
S ζ ζ real number line S Eample A random eperimen of ossing 3 fair coins Sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Le be he number of heads; hen S {,,, 3} eg 3 3 THH ; P[ ], P[ ], P[ ], P[ 3]
Equivalen evens Le A be he se of oucomes ζ in S ha leads o he se of values ζ in B A B eg A B { ζ S : ζ B} in he above coins ossing eample, {, 3} {HHT, HTH, THH, HHH} se of all preimages of elemens in B {, 3}
Since even B in S occurs whenever even A in S occurs, and vice versa Hence P[B] P[A] P[{ζ: ζ in B}] A and B are called equivalen evens wih respec o If we assign probabiliies in his manner, hen he probabiliies assigned o subses of he real line will saisfy he hree aioms of probabiliy P[B] for all B S P[S ] 3 If B and B are muually eclusive, hen P[B B ] P [B ] + P[B ] In he ossing coins eperimen, we observe 7 P[ ], P[ ], P[ ], P[ 3] Hence, P[ ] is a number whose value depends on, and so i is a funcion of
Eample Consider he random eperimen of ossing 3 coins S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} no of heads in he 3 coins, S {,,, 3} A {HTT, TTT} A {HHT, HTH, THH, TTT} {HTT, THT, TTH, TTT} A 3 A {, } se of all values aken by ζ, ζ A A {, } {, } se of all preimages of elemens in {, } {HTT, THT, TTH, TTT} A 3
Noe ha A 3 and {, } are equivalen evens since Noe ha A 3 S and {, } S Consider anoher random variable: P[ A3 ] P[ or ] Y number of heads number of ails hen Y can assume he values 3,, and 3 Now, Y {3, } {TTT, HTT, THT, TTH}, so {TTT, HTT, THT, TTH}, and {3, } are equivalen evens
Eample A poin is seleced a random from inside he uni circle cenered a he origin Le Y be he random variable represening he disance of he poin from he origin a S Y {y: y } range of Y y cener seleced poin b The equivalen even in he sample space S for he even {Y y} in S Y is ha he seleced poin falls inside he region cenered a he origin and wih radius y c P[Y y] y probabiliy of selecing a poin inside he uni circle, and whose disance is less han or equal o y πy y π
Le Z be he random variable represening he disance of he seleced poin from, a 3 z : z S Z, z y The equivalen even in S for he even {Z z} is he region formed by he inersecion of he circles: + + z y y b
Cumulaive disribuion funcion cdf The cdf of a random variable is defined as P[ ], < < Aioms of probabiliy following properies of cdf lim sure even 3 lim impossible even 4 is a non-decreasing funcion of This is obvious since for >, we have P[ ] P[ ]
5 is coninuous from he righ ie for h > lim + + + b h b b h Eample The ossing coins eperimen again, where number of heads appearing in ossing 3 coins Take h > and h +, ] [ head} or { ] [ head} { ] [ + + h P h P P P h P h
Hence, he cdf of is coninuous from he righ Define he uni sep funcion: 7 u <, 3 3 3 u + u + u + u 3 The jump a is given by P[ ], and similarly, for he jump a, and 3
6 P[a < b] b a since { a} {a < b} { b}, and { a} and {a < b} are muually eclusive so a + P[a < b] b Suppose we ake a b h, h >, P[b h < b] b b h As h +, P[ b] b b The probabiliy ha akes on he special value b is given by he magniude of he jump of he cdf a b h If he cdf is coninuous a b, hen he even { b} has probabiliy zero essenially h If he cdf is coninuous a a and b, hen P[a < < b], P[a < b], P[a < b], P[a b] have he same value 7 P[ > ]
Eample Le T be he random variable which equals he life of a diode Suppose he cdf of T akes he form, ] [ - T e u e - P T µ µ < hen he probabiliy ha he diode fails beween imes a and b is P[a < T b] P[T b] P[T a] e µa e µb T
Three ypes of random variables Discree random variable The cdf is a righ-coninuous, saircase funcion of wih jumps a a counable se of poins,, P u k where P k P[ k ] gives he magniude of he jump a k in he cdf Coninuous random variable The cdf is coninuous everywhere, so P[ ] for all k k
3 Random variable of mied ype The cdf has jumps on a counable se of poins and also increases coninuously over a leas one inerval of values of p + p, < p < cdf of a discree random variable cdf of a coninuous random variable
Eample Le be he ime insan ha a cusomer in a queue is being served We have: is zero if he sysem is idle and eponenially disribued if he sysem is busy P[ ] P[ idle] P[idle] + P[ busy] P[busy] p probabiliy ha he sysem is idle p idle + p busy p + pu + p e λ p u e < λ p idle is a discree random variable wih P[ idle ] so ha idle u; busy is coninuous wih busy u - e λ
Probabiliy densiy funcion pdf, if eiss, is defined as d d f f + P[ < + ] f since ] [ P + + + <
Properies of pdf f since cdf is a non-decreasing funcion of f d d Proof: rom f, we obain d f d The consan c is deermined by, given c c 3 P [ a < Proof: b] P[ a < b a f d b] b a 4 d f b f d f d a b a f d
Eample Le radius of bull-eye b and radius of arge a Probabiliy of he dar sriking beween r and r + dr is ] [ dr a r C dr r R r P + R disance of hi from he cener of he arge a b The densiy funcion akes he form a r C r f R Assume ha he arge is always hi:, 3 a C dr a r C a probabiliy of hiing bull-eye 3 ] [ 3 a b a b dr r f b R P b R How o deermine C?
pdf for a discree random variable The dela funcion δ is relaed o u via d δ u or u δ d Noe ha δ d d Recall ha P u k k k probabiliy mass funcion According o f d, we hen have when k f P k δ k, where δ k k oherwise u k δ k k k
Eample The coins ossing eperimen cdf: pdf: f 3 3 u + u + u + u 3 3 3 δ + δ + δ + δ 3 P[ < ] + f d P[ ] 3 Noe ha he dela funcion locaed a is ecluded bu he dela funcion locaed a is included Similarly, [ < 3] P f d P[ 3 ] 3
Condiional cdf of given A P[{ } ] A A if P[ A] > P[ A] h cdf of wih reference o he reduced sample space A Condiional pdf of given A f d A A d S { } A
Eample The lifeime of a machine has a coninuous cdf, ind he condiional cdf and pdf given he even A { > }, ha is, he machine is sill working a ime Condiional cdf > P[ > ] P[{ } { P[ > ] > }] i ii > { > } { } { > } { }
Noe ha, } { } { } { > < > φ > > so Condiional pdf is found by differeniaing wih respec o > > > f f Noe ha > is coninuous a, bu f > has a jump a
Eample Tossing of 3 coins; number of heads; A { > } 3 if 3 if if ] [ ] [ if ] [ }] { } [{ < > < > > > P P P P A A 3 Noe ha P[ > ] P[ 3] and 3 if 3 if < P[ < ]