Definition A continuous random variable X on a probability space (Ω, F, P) is a function X : Ω R such that for all x R. 1 {X x} is an event, and

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1 Continuous random variabls I Mathmatics for Informatics 4a José Figuroa-O Farrill Lctur 7 Fbruary 22 Aftr discrt random variabls, it is now tim to study continuous random variabls; namly, thos taking valus in an uncountabl st,.g., R For xampl, choos at random a ral numbr btwn and. What is th probability of choosing 7? At random mans that vry numbr is qually likly, so th probability of choosing 7 is th sam as that of any othr numbr. Lt s call that proability ɛ. What can ɛ b? W can writ th crtain vnt [, ] as th disjoint union [, ] {x} x [,] W know that P([, ]), but this is not a countabl disjoint union. José Figuroa-O Farrill mi4a (Probability) Lctur / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 3 / 25 Continuous random variabls II So lt us brak up [, ] into a countabl disjoint union: [, ] A { n } n whr A is simply th complmnt of {, 2, 3,... }. Assuming that { n } is an vnt for all n, w apply P to obtain P(A ) + ɛ ɛ n This shows, by th way, why on limits th additivity of P to countabl unions; othrwis on would conclud that P([, ]) a contradiction. This argumnt also shows that any countabl subst of R has zro probability: rationals, algbraic numbrs,... José Figuroa-O Farrill mi4a (Probability) Lctur 4 / 25 Continuous random variabls III Dfinition A continuous random variabl X on a probability spac (Ω, F, P) is a function X : Ω R such that for all x R {X x} is an vnt, and 2 P(X x) Rmark Th dfinition rquirs {X x} to b an vnt. Lt s prov it. Lt B n {X x n } for n, 2,.... Thy ar vnts, whnc so ar thir union n B n {X < x}, its complmnt {X x}, and finally thir intrsction {X x} {X x} {X x}. José Figuroa-O Farrill mi4a (Probability) Lctur 5 / 25

2 Exampl Th probability spac modlling th motivating xampl of choosing a numbr at random in [, ], is thn th tripl (Ω, F, P), whr Ω [, ]; 2 F consists of th intrvals [, a] with a togthr with and any othr substs thy gnrat by itrating complmntation and countabl unions; and 3 P([, a]) a. Dfinition Th distribution function F of a continuous random variabl X is th function P(X x). In th abov xampl, x. Probability dnsity functions In this cours w will b daling xclusivly with continuous random variabls whos distribution function F is givn by intgrating a function f: f(y)dy. Th function f is calld a probability dnsity function (p.d.f.) and th function F is calld a cumulativ distribution function (c.d.f.). José Figuroa-O Farrill mi4a (Probability) Lctur 6 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 7 / 25 PDFs and CDFs Dfinition A probability dnsity function is a function f(x) normalisd such that f(x)dx Givn f, th non-dcrasing function F dfind by f(y)dy is calld th cumulativ distribution function of f. Discrt random variabls hav probability mass functions, but continuous random variabls hav probability dnsity functions. Continuous random variabls and PDFs As with discrt random variabls, w oftn just say Lt X b a continuous random variabl with probability dnsity function f(x)... without spcifying th probability spac on which X is dfind. Th basic proprty of th probability dnsity function for a continuous random variabl X is that P(X A) f(x)dx assuming that {X A} is an vnt. This prompts th following Qustion x A For which substs A R is {X A} an vnt? José Figuroa-O Farrill mi4a (Probability) Lctur 8 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 9 / 25

3 Borl sts Such substs ar calld Borl sts. By dfinition, (, x] is a Borl st for all x R. So ar (, x) n (, x n ]. By complmntation, so ar (x, ) and [x, ) By intrsction, [x, y] (, y] [x, ) and similarly (x, y), [x, y), (x, y],... Th Borl sts ar th smallst σ-fild containing th intrvals. In fact, all substs of R you will vr b likly to mt ar Borl. Proprtis of cumulativ distribution functions Lt f b a probability dnsity function with cumulativ distribution function F. Rmmbr that f(y)dy f(x). Thn F satisfis th following proprtis: F() and F( ) if x y, thn F(x) F(y) F (x) f(x) F(b) F(a) b a f(x)dx José Figuroa-O Farrill mi4a (Probability) Lctur / 25 José Figuroa-O Farrill mi4a (Probability) Lctur / 25 Exampl (Th uniform distribution) Th p.d.f. of th uniform distribution on [a, b] is givn by, x < a f(x) b a, a x b, x > b and th c.d.f. is givn by, x < a x a b a, a x b, x > b Exampl (Waiting for th bus) Btwn 4pm and 5pm, buss arriv at your stop at 4pm and thn vry 5 minuts until 5pm. You arriv at th stop at a random tim btwn 4pm and 5pm. What is th probability that you will hav to wait at last 5 minuts for th bus? Your arrival tim at th stop is uniformly distributd btwn 4pm and 5pm. You will hav to wait at last 5 minuts if you arriv btwn th tim a bus arrivs and minuts aftr that. That s minuts in vry 5 minuts, so th probability is José Figuroa-O Farrill mi4a (Probability) Lctur 2 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 3 / 25

4 Exampl (Th standard normal distribution) Exampl (Th standard normal distribution continud) Th p.d.f. of th standard normal distribution is Th proof that f(x) is a probability dnsity function follows from a standard trick. Lt x2 /2 f(x) x I It is also calld a gaussian distribution. 2 /2 dx which w hav to show to b qual to. W comput I2 : 2 x2 /2 dx I y2 /2 x2 /2 dx dy x (x2 +y2 )/2 dx dy 2 whr th intgral is ovr th whol (x, y)-plan. W now chang to polar coordinats. Jos Figuroa-O Farrill mi4a (Probability) Lctur 4 / 25 Exampl (Th standard normal distribution continud) x r cos θ y r sin θ r> mi4a (Probability) Lctur 5 / 25 Exampl (Th normal distribution) Th normal distribution with paramtrs µ and σ2 has probability dnsity function 6 θ < so that x2 + y2 r2 Jos Figuroa-O Farrill dx dy r dr dθ f(x) σ Into I2, x (x2 +y2 )/2 dx dy r2 /2 r dr dθ θ r (x µ)2 2σ2 I2 2 /2 r Th standard normal distribution has µ and σ. W will s that µ and σ2 ar th man and varianc, rspctivly. d( 2 r2 ) u du Jos Figuroa-O Farrill mi4a (Probability) Lctur 6 / 25 Jos Figuroa-O Farrill mi4a (Probability) Lctur 7 / 25

5 Exampl (Th rror function) Th cumulativ distribution function of th normal distribution is ( ) x µ rf 2σ whr rf is th rror function, dfind by Exampl (Th xponntial distribution) Th p.d.f. of th xponntial distribution with paramtr λ is { λ λx, x f(x), x < rf(x) 2 t2 dt π and th c.d.f. is givn by { λx, x, x < José Figuroa-O Farrill mi4a (Probability) Lctur 8 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 9 / 25 Th xponntial distribution has no mmory Lt X b xponntially distributd with paramtr λ.thn P(X x) λx for x, whnc P(X > x) P(X x) λx If x, y, from λ(x+y) λx λy w hav that P(X > x + y) P(X > x)p(x > y) or quivalntly, partitioning Ω {X > x} {X x}, P(X > x + y) P(X > x + y X > x)p(x > x) + P(X > x + y X x)p(x x) whnc canclling th P(X > x) from both sids, Exampl Lt X dnot th tim it taks for a computr programm to crash It is snsibl to assum that X is xponntially distributd Th conditional probability of it not crashing aftr a tim x + y givn that it didn t crash aftr a tim x is P(X > x + y X > x) which was shown to qual P(X > y) which is th probability of not crashing aftr a tim y so th fact that it didn t crash for a tim x is of no rlvanc i.., th xponntial distribution simply dos not rmmbr that fact. P(X > x + y X > x) P(X > y) José Figuroa-O Farrill mi4a (Probability) Lctur 2 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 2 / 25

6 Expctations Lt X b a continuous random variabl with probability dnsity function f(x). Thn w dfin its xpctation (or man) by E(X) xf(x)dx (providd th intgral xists) Notic that if f is symmtric, so that f( x) f(x), thn E(X). Exampl (Th man of th xponntial distribution) Lt X b xponntially distributd with paramtr λ. Thn E(X) λ d dλ xλ λx dx λ x λx dx λx dx λ d dλ λ λ Exampl (Th man of th uniform distribution) Lt X b uniformly distributd in [a, b]. Thn E(X) b a x b a dx 2(b a) x2 b a b2 a 2 2(b a) b + a 2 In othr words, th man is th midpoint in th intrval. Exampl (Waiting for th bus continud) In th xampl about waiting for th bus, what is your xpctd waiting tim? Your xpctd arrival tim is uniformly distributd, but you ar intrstd in th xpctatation of th waiting tim. Bcaus of th priodicity of th buss, it is nough to considr a 5-minut intrval: 4pm-4:5pm, say. Thn th waiting tim is th sam as th arrival tim, and hnc th xpctation is th midpoint of th intrval, so 7 2 minuts. José Figuroa-O Farrill mi4a (Probability) Lctur 22 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 23 / 25 Exampl (Th man of th normal distribution) Lt X b normally distributd with paramtrs µ and σ 2. Thn E(X) σ x (x µ)2 /2σ 2 dx W chang coordinats to y x µ, so that E(X) σ σ + µ µ (y + µ) y2 /2σ 2 dy y y2 /2σ 2 dy + σ µ y2 /2σ 2 dy whr th first trm vanishs bcaus of symmtric intgration and th scond quals µ by using th normalisation of th normal probability dnsity function. Summary X a continuous random variabl: for all x, {X x} is an vnt and P(X x) Continuous random variabls hav continuous distribution functions P(X x) F is oftn dfind by a probability dnsity function f: f(y)dy f(x) f(x)dx and is calld a cumulativ distribution function W hav mt svral probability dnsity functions: uniform: f(x) b a for x [a, b] normal: f(x) σ /2σ 2 (x µ)2 xponntial: f(x) λ λx for x (has no mmory!) Th man µ a+b xf(x)dx, and quals 2, µ and λ for th abov p.d.f.s, rspctivly. José Figuroa-O Farrill mi4a (Probability) Lctur 24 / 25 José Figuroa-O Farrill mi4a (Probability) Lctur 25 / 25