Probabiliy The probabiliy PrA) of an even A is a number in [, ] ha represens how likely A is o occur. The larger he value of PrA), he more likely he even is o occur. PrA) means he even mus occur. PrA) means he even canno occur. PrA) > PrB) means A is more likely o occur han B. Evens A and B are called disjoin muually exclusive) if hey canno boh occur simulaneously, ha is, if PrA and B). Equivalenly, saying A and B are disjoin means PrA or B) PrA) + PrB). Evens A and B are called complemenary if hey are disjoin, bu one of hem mus occur. Equivalenly, PrA) + PrB) PrA or B). An random variable is a quaniy ha assumes differen values wih cerain probabiliies. In oher words, X is a random variable if we can assign values o PrX x), PrX x), PrX < x), PrX x), PrX > x) for every real number x. The evens X x and X x are complimenary: PrX x) PrX x). The evens X x and X > x are complimenary: PrX x) PrX > x). The evens X x and X < x are complimenary: PrX x) PrX < x). Example. If we oss wo fair coins, here are four possible oucomes: HH HT TH TT where H is heads and T is ails. Since he coins are fair and he osses are independen he oucome of one oss doesn affec he oucome of he oher), each of he four oucomes has probabiliy 2 2 4. Le Y be he random variable defined by if he oucome is HH if he oucome is HT Y 2 if he oucome is TH if he oucome is TT
2 Then PrY k) for k,, 2, 4 PrY ) PrY π) PrY > 2) PrY ) 4 PrY 2) PrY or Y or Y 2) PrY ) + PrY ) + PrY 2) 4 PrY 2) PrY > 2) 4 4 The random variable Y is an example of a discree random variable. A random variable is called a discree random variable if i assume only finiely many or counably many values, ha is, if we can lis he values i assumes as x, x 2,...,. Roughly, a coninuous random variable is one ha can assume a coninuum of values. We will give a precise definiion below afer saing some erminology, bu here are some hings ha can be modelled by a coninuous random variables: he amoun of rainfall in vancouver nex week he lifespan of a lighbulb he heigh of a randomly seleced person in Canada The cumulaive disribuion funcion CDF) of a random variable X is he funcion F x) PrX x). A funcion F is a CDF of some random variable if and only if he following properies hold. ) F is righ-coninuous: lim x c + F x) F c) for all real numbers c. 2) F is non-decreasing: F x) F y) when x y for all real numbers x, y. ) lim F x). x 4) lim F x). x Properies 2), ), and 4) imply F x) for all real numbers x. Example. Le a > be a consan. Show ha F x) funcion. is a cumulaive disribuion + e ax ) Since + e ax is coninuous and never, F x) is coninuous and, herefore, righ-coninuous everywhere. 2) Since d dx F x) d dx + e ax ) + e ax ) 2 ae ax ) for all x, F is non-decreasing. ) lim F x) lim x x + e ax + e ax + e ax ) 2,
4) lim x F x) Therefore F x) lim x + e ax is a CDF. + e ax Exra Example. The funcion F x) k arcanx) + 2 is a cumulaive disribuion funcion. Find he value of k. We mus have lim F x). So x Solving for k yields k π. lim F x) lim x x k arcanx) + ) k π 2 2 + 2 A random variable is called a coninuous random variable if is CDF is coninuous. If X is a coninuous random variable, hen PrX x), PrX x) PrX < x), PrX x) PrX > x), Pra X b) Pra < X b) Pra X < b) Pra < X < b). for all real numbers x, a, b. We will prove PrX x) below; all he oher formulas follow easily from i. For a coninuous random variable X whose cumulaive disribuion funcion F is differeniable, he probabiliy densiy funcion PDF) of X is defined o be and, moreover, for all real numbers x, a, b wih a b. fx) d dx F x), PrX x) F x) Pra X b) F b) F a) f) d b a fx) dx A funcion f is a PDF of some random variable if and only if he following properies hold. ) f is non-negaive: fx) for all real numbers x. 2) Example. Le fx) dx fx) { kx x) 2 if x oherwise be he probabiliy disribuion funcion of a random variable X. a) Find he value of he consan k. b) Find he probabiliy ha 2 X <. c) Find he cumulaive disribuion funcion of X.
4 Par a): We mus have fx) dx. Therefore k k fx) dx kx x) 2 dx x 2x 2 + x ) dx k ) k 2. 2 2 + 4 2 x2 2 x + ) 4 x4 So k 2. Par b): ) Pr 2 X < /2 fx) dx 2 /2 x x) 2 dx 2 x 2x 2 + x ) dx 2 /2 2 x2 2 x + ) 4 x4 [ 2 2 )2 2 ) + ) ) 2 4 )4 2 2 2 2 [ 2 2 8 2 + 64 )] 5 6. /2 ) + 4 ) )] 4 2 Par c). For x <, F x) PrX x) f) d d. For x, F x) PrX x) f) d 2 ) 2 d 2 2 2 + ) d 2 2 2 2 + ) x 4 4 2 2 x2 2 x + ) 4 x4.
5 For x >, F x) PrX x) 2 2 f) d + d + 2 f) d f) d + ) 2 d + 2 2 + ) d ). 2 2 + 4 f) d d 2 2 2 + 4 4 ) Par b) again: Remember ha Pra X b) F b) F a) b a fx) dx, where F is he CDF of X and f is he PDF of X. So if we had done par c) before par b), we could have used ha if x < F x) 2 2 x2 2 x + 4 x4) if x if x > o compue ) ) Pr 2 X < F ) F 2 2 2 )2 2 ) + ) ) 2 4 )4 2 2 ) + ) ) 4 2 2 2 4 2 5 6. Example. Find he probabiliy disribuion funcion fx) for a random variable wih cumulaive disribuion funcion { F x) e 2 x if x if x < For x >, For x <, fx) d dx F x) d e x) 2 dx 2 e 2 x. fx) d dx F x) d dx. F x) is no differeniable a, bu i s okay o have fx) undefined a finiely many poins because his won affec he inergal of f. In conclusion, { fx) 2 e 2 x if x > if x <.
6 Le X be a coninuous random variable wih probabiliy densiy funcion f. The expeced value or expecaion or mean or average) of X is The variance of X is VarX) EX) xfx) dx. x EX)) 2 fx) dx EX 2 ) EX)) 2 The sandard deviaion of X is σx) VarX). Here is a useful fac see below for a proof): EX 2 x 2 fx) dx. The expeced value of a random variable is a measure of he cener of is disribuion. The variance and sandard deviaion of a random variable are measures of he dispersion or horizonal spread) of is disribuion. Example. Find he sandard deviaion of he random variable X wih probabiliy densiy funcion fx) x + ) 4 if x if x < Firs we find he expeced value: EX) xfx) dx xx + ) 4 dx + + xx + ) 4 dx u )u 4 du u x + u u 4 ) du 2 u 2 + ) + u du 2 + ) 2 + ) + ) + + 2 ) 2. 2 + ))
7 Then we compue EX 2 ) x 2 fx) dx x 2 x + ) 4 dx + + x 2 x + ) 4 dx u ) 2 u 4 du u x + u 2 2u + u 4 ) du u + u 2 ) + u du + ) + + ) 2 ) + ) + + ) + )) Therefore and. VarX) EX 2 ) EX)) 2 σx) VarX) 2. ) 2 2 4
8 Proofs of Two Facs Fac. If X is a coninuous random variable, hen PrX c) for all real numbers c. If X has a probabiliy densiy funcion, he proof is easy: PrX c) Prc X c) c c fx) dx. If X is a coninuous random variable wihou a probabiliy densiy funcion, we have o work harder. We will need o recall he definiion of a coninuous random variable and a propery of limis. Recall: X is a coninuous random variable if is CDF, F x) PrX x), is coninuous. Recall: If m gx) for all x < c, hen m lim x c fx). Proof of Fac. Sep. For any x < c, and herefore PrX c) PrX x or x < X c) PrX x) + Prx < X c) Prx < X c) PrX c) PrX x) F c) F x). Sep 2. For any x < c, if X c, hen x < X c. So X c canno be more likely han x < X c. Therefore PrX c) Prx < X c). Sep. PrX c) lim PrX c) x c lim Prx < X c) x c lim F c) F x)) x c F c) lim F x) x c F c) F c).
9 Fac 2. If X is a coninuous random variable wih probabiliy densiy funcion f, hen EX 2 ) x 2 fx) dx. Remark. Fac 2 is a special case of a more general formula called he law of he unconscious saisician, EgX)) gx)fx) dx, which is valid whenever he inegral on he righ converges. Proof of Fac 2. Le F and f be, respecively, he CDF and PDF of X. Le G and g be, respecively, he CDF and PDF of he random variable X 2. If x <, If x, Gx) PrX 2 x). Gx) PrX 2 x) Pr X x) Pr x X x) F x) F x). Now we compue gx). If x <, If x >, gx) d dx Gx) d dx. gx) d dx Gx) d ) F x) F x) dx F x) d dx x) F x) d dx x) F x) 2 x + F x) 2 x 2 ) f x) + f x). x We don need o worry abou gx) a x, since he value of g a one poin won affec he inegral of g. We can now compue Making he subsiuion EX 2 ) 2 2 lim xgx) dx x 2 ) f x) + f x) dx x ) x f x) + f x) dx x f x) + f x) ) dx. u x, u 2 x, 2u du dx x u, x u
gives EX 2 ) lim u 2 fu) + f u)) du ) lim u 2 fu) du + u 2 f u) du. In he second inegral we make he subsiuion v u o ge EX 2 ) lim u 2 fu) du lim lim v 2 fv) dv u 2 fu) du + v 2 fv) dv u 2 fu) du + lim u 2 fu) du + x 2 fx) dx + x 2 fx) dx. v 2 fv) dv x 2 fx) dx ) ) v 2 fv) dv