Ch.3 Random Variables 불규칙변수 3. The Noion of a Random Variable S ζ ζ real line Random variable a funcion ha assigns a real number ζ o each oucome ζ in he sample space of a random eperimen Specificaion of a measuremen on he oucome of a random eperimen Define a funcion on he sample space, i.e., a random variable S
S he domain of he random variable S he range of he random variable E.3.: Afer hree ime of coin ossing, he sequence en e of heads and ails is noed. S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} ζ oucome he number of heads in hree coin osses S {0,,, 3} a new sample space
ζ : HHH HHT HTH THH HTT THT TTH TTT ζ : 3 0 cf If ζ is already a numerical value, he oucome can be a random variable defined by ideniy funcion, ζ ζ uncion is fied and deerminisic Randomness is due o he eperimen oucomes ζ Sample space S, random oucome ζ and he even A Sample space S,, random variable ζ and he even B in S, A {ζ : ζ in B} } B A {ζ : ζ in B} 3
3.. Cumulaive Disribuion uncion cdf The cdf of a random variable : he probabiliy of he even { } } for - < < a random variable a numerical variable no random The probabiliy pobab ha aes on a value au in he se -, The probabiliy of he even {ζ : ζ } Noe The even { } and vary wih or : a funcion of he variable. 4
Evens of ineres when dealing wih numbers Inervals of he real line, heir complemens, unions, inersecions roperies of he CD 0 cf Aiom I : 0 A and Corollary : A lim cf The even { < } is he enire sample space, hen Aiom II : S 5
lim 0 cf The even { - } is he empy se, hen Corollary 3: Φ0 is a nondecreasing funcion of, i.e., if a < b, hen a b 단조증가함수 cf Corollary 7: he even { a} he even { b}, hen a b is coninuous from he righ + for h >0 0, b lim b + h b h 0 6
a < b b- a cf { a} {a < b} { b}, hen by Aiom III a + a < b b b b- - b cf le a b-ε and ε > 0 b-ε < b b- b-ε lim b ε < ε 0 b b b b Noe : If he CD is coninuous a a poin b, hen he even { b} has probabiliy zero, i.e., b 0 if CD is coninuous a a poin b. 7
Noe : {a b} { a} {a < b} a b a- a - + b- a b- a - Noe : If he CD is coninuous a endpoins, i.e., a he poins a and b, hen a < < b a < b a < b a b > - 8
E 3.4: The cdf is coninuous from he righ and equal o ½ a he poin. 3/8, he magniude of he jump a he poin. cf or a small posiive number δ, δ,, + δ 8 3 δ 8 0 < 0 cf In erms of uni sep funcion u 0 3 3 u + u + u + u 3 8 8 8 8 9
Three Types of Random Variables Discree Random Variable cdf is a righ-coninuous, saircase funcion of wih jumps a a counable se of poins 0,,, S { 0,,, } probabiliy mass funcion pmf of : he se of probabiliies p of he elemens in S cdf of a discree random variable p u where p : he magniude of jumps in he cdf 0
Coninuous Random Variable is coninuous everywhere and sufficienly smooh. f d where f is a nonnegaive funcion 0
Random Variable of Mied Type cdf has jumps on a counable se of poins 0 0,,,, cdf increases coninuously over a leas one inerval of values of. p + -p where 0<p<, and is he cdf of a discree random variable and is he cdf of a coninuous random variable
3.3. The robabiliy Densiy uncion pdf The probabiliy densiy funcion of pdf : f d f Noe : The pdf represens he densiy of probabiliy a d f Noe : The pdf represens he densiy of probabiliy a he poin. h h + + < h h h + li. f h h h + where inerval for very small 3 lim 0 f h h where
f 0 cf d f and is a nondecreasing funcion d < + d f d The probabiliies of evens of fhe form. falls in a small Inerval of widh d abou he poin 4
a b f d b a cf The probabiliy of an inerval a, b The area under f in ha inerval b a a b f d 5
3 f d Noe : The pdf compleely specifies he behavior of coninuous random variables 4 d f : normalizaion condiion for pdf s Noe : a valid pdf can be formed from any nonnegaive, piecewise i coninuous funcion g, ha has a finie i inegral g d c < g f c 6
E. 3.8 The pdf of he samples of he ampliude of speech waveforms. f ce α < < c? using he normalizaion condiion c α α c ce d α c 7
α v < v e v α e αv α v 0 d e α d 8
pdf of Discree Random Variables cf The derivaive of he cdf does no eis a he disconinuiies. cf The relaion beween u and δ Uni sep funcion 0 < 0 u 0 Dela funcion u δ d definiion of he dela funcion 9
The cdf of a discree random variable u p p where Generalized definiion of he pdf f d f b u b f δ p f δ 0 The pdf for a discree random variable
E.3.9 : he number of heads in hree coin osses. S {0,,, 3} ind he pdf of ind < and < 3 by inegraing he pdf. sol 3 3 u + u + u + u 3 8 8 8 8 3 3 f δ + δ + δ + δ 3 8 8 8 8 + 3 < f d + 3 8 3 3 < 3 f d 8
Definiion of Condiional cdf s and pdf s condiional cdf of given A { } A A A if A > condiional pdf of given A d f A A d E. 3.0 lf life-ime of a machine : random variable cdf ind he condiional cdf and pdf given he even A { > } i.e., machine is sill woring a ime 0
sol The condiional cdf > > } { } { > > } { } { } { } { } { < > > < φ if 0 > } { } { } { cf coninuous a > > 3 cf coninuous a
The condiional pdf f f f f > H W 4 9 3 5 6 7 3 6 7 H.W., 4, 9, 3, 5, 6, 7, 3, 6, 7 4
3.4. Some Imporan Random Variables Discree Random Variables Bernoulli Random Variable Binomial Random Variable Geomeric Random Variable Negaive Binomial Random Variable oisson Random Variable 5
Coninuous Random Variable Uniform Random Variable Eponenial Random Variable Gaussian Random Variable m-erlang Random Variable Chi-Square Random Variable Rayleigh Random Variable Cauchy Random Variable Laplcian Random Variable 6
Discree Random Variables - Couning is involved Bernoulli Random Variable 0 if ζ no in A I A ζ if ζ in A Indicaor funcion for A I A Random Variable S {0, } pmf robabiliy Mass uncion p I 0 -p and p I p where A p cf ossing a biased coin I A : Bernoulli random variable 7
Binomial Random Variable n imes of independen rials I j : The indicaor funcion for he even A in he j h rial j I +I + + I n cf I j 0 or pmf of n n p p for : binomial random variable 0,,n Noe : rom he pdf, is maimum a ma n+p, where is he larges ineger ha is smaller han or equal o Noe : If n+p is an ineger, hen he maimum is achieved a ma and ma - rob. 33 8
Geomeric Random Variable The number M of independen Bernoulli rials unil he firs occurrence of a success S M {,, } pmf of M M -p - p,, where p A is he probabiliy pobab yof success in each Bernoulli rial rom he pdf M decays geomerically wih. cf q - p 9
cdf j q q p pq M cf Geomeric Series j q q p pq n a r n a r a ar ar r + + + Memoryless ropery, j M j M j M j M j M + > + > + M j M j M j M j M > > > + 30
cf M q j M j j + + M M q pq M j j < < q j M q j + + M q q q j M j M j + + 3
The only memoryless discree random variable cf memoryless : Each ime a failure occurs, he sysem forges and begins anew as if i were performing he firs rial e.g. The memoryless propery M + j M > j M for all j, > 3
oisson Random Variable Couning he number of occurrences of an even in a cerain ime period or in a cerain region in space. where he evens occur compleely a random in ime or space Couns of emissions from radioacive subsances Couns of demands for call connecion Couns of defecs in a semiconducor chip pmf for he oisson random variable α α N e 0,,,,,,! where α is he average number of even occurrences in a specified ime inerval or region in space. 33
Noe for α <, N is maimum a 0 for α >, N is maimum a α Noe if α is a posiive ineger, ma α and α- α α α α α α e e e e! 0! 0 cf! α! α α lim 0! even for very large α 34
Approimaion of he binomial probabiliies wih very large n and small p. cf p n p α! n α p e where α np 3 n α α α n α e α + + + +! 3! n! n α α α p0 n+ n n + n n! n α + α + e α! p + n p / n α α p + p + α / n + 35
f f cf + +!! f a a f a a f a f f n + +! h h a n a f n n + + + +!! a f h a f h a f h a f 36
E. 3. Rae of requess for elephone connecions λ calls/sec The number of requess in a ime period oisson random variable ind he probabiliy bili for no call requess in seconds ind he probabiliy for n or more requess 37
sol Average number of requess in seconds α λ e e N λ λ λ 0! 0 0 n n N n N λ λ < e λ λ 0! 38
Coninuous Random Variables Easier o handle analyically The limiing form of many discree random variables coninuous random variable Uniform Random Variable All values in an inerval of he real line are equally liely o occur 39
Eponenial Random Variable The ime beween occurrence of evens The lifeime of devices and sysems pdf and cdf f 0 λe < 0 0 e λ λ 0 < 0 0 f λe -λ -e -λ 40
λ he rae a which evens occur cf he probabiliy of an even occurring by ime increases as he rae λ increases. Limiing form of he geomeric random variable An inerval of duraion T Subinervals of lengh T cf n T n,, 0 discree model coninuous model n The sequence of subinervals a sequence of independen α Bernoulli rials wih p n where α he average number of evens per T seconds. 4
The number of subinervals unil he occurrence of an even a geomeric random variable M. T The ime unil he occurrence of he firs even n T M n M > > n M T T n p T T n p p 4 T p
n T α α e T as n n e α T Noe : oisson random variable N 0 e -λ No calls for seconds he eponenial random variable α cdf wih λ T an inerval beween any wo calls > seconds 43
Meanwhile α T > e : probabiliy ha > seconds In conclusion, for oisson random variable. he ime beween evens is an eponenially disribued random α variable wih λ T Noe : The eponenial random variable is he only coninuous random variable ha saisfies he memoryless propery. 44
GaussianNormal Random Variable A random variable consising of he sum of a large number of small random variables approaches he Gaussian random variable : he cenral limi heorem The pdf for he Gaussian random variable m σ f e < < πσ Where m and σ are he mean and sandard deviaion of. 45
e πσ cdf m σ πσ Φ m σ m σ where Φ e d π cf Φ is he cdf of a Gaussian random variable wih m 0 and σ e d d 46
E. 3.4: Show ha e d π σ σ m m0 47
y e d e d e dy π π y e + ddy π π r e rdrdθ π 0 0 e π d cf e d π 48
Q-funcion Q Φ d e Q Φ π he probabiliy of he ail of he pdf π, 0 Q Q Q e Q π π, + + b a b a a Q where 49 π π, b a where
Noe : Table 3.4 The value of for which Q 0 - where,,,0 : he probabiliy of ale 0. or : he probabiliy of ale 0.0 0. or 0.0-0 "Zero mean Gaussian 50
pdf Gamma Random Variable α λ λ λ e f 0 < Γ α < where wo parameers α and λ are posiive numbers and Γz is he gamma funcion. cf Γ z 0 z Γ m + m! e Γ π Γ z + zγ z d for for z for m > z > 0 0 a nonnegaive ineger 5
Many random variables are special cases of he gamma random variable The eponenial random variable Special case of he gamma r.v. wih α 5
Chi-Square Random Variable λ, α where is a posiive ineger f e Γ e Γ 53
m-erlang Random Variable α m m λ e λ λ f > 0 m! The ime S m ha elapses unil he occurrence of he m h even. : m-erlang r.v. S m + + + m,,, m : he imes beween evens eponenial random variables S m if and only if m or more evens occur in seconds N m m h even occurred by ime 54
m N m N S m S m < where N is he oisson random variable for he number of evens in seconds. cf α λ d d f e m m m S S m S,! 0 λ λ So, S m is an m-erlang random variable! 55