Lecture 4. Random variable and distribution of probability
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1 Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH Itroductio to probability ad statistics, Lecture
2 Outlie: Cocept of radom variable Quatitative descriptio of radom variables Examples of probability distributios Itroductio to probability ad statistics, Lecture
3 The cocept of radom variable Radom variable is a fuctio X, that attributes a real value x to a certai results of a radom experimet. Ω { e, e, X : Ω R X( e ) x i i } K R Examples: ) Coi toss: evet head taes a value of ; evet tails - 0. ) Products: evet failure - 0, well-performig 3) Dice:, etc. ) Iterval [a, b] a choice of a poit of a coordiate x is attributed a value, e.g. si (3x+7) etc.. Itroductio to probability ad statistics, Lecture 3
4 The cocept of radom variable Radom variable Discrete Whe the values of radom variable X are isolated poits o a umber lie Toss of a coi Trasmissio errors Faulty elemets o a productio lie A umber of coectios comig i 5 miutes Cotiuous Whe the values of radom variable cover all poits of a iterval Electrical curret, I Temperature, T Pressure, p Itroductio to probability ad statistics, Lecture
5 Quatitative descriptio of radom variables Probability distributios ad probability mass fuctios (for discrete radom variables) Probability desity fuctios (for cotiuous variables) Cumulative distributio fuctio (distributio fuctio for discrete ad cotiuous variables) Characteristic quatities (expected value, variace, quatiles, etc.) Itroductio to probability ad statistics, Lecture 5
6 Distributio of radom variable Distributio of radom variable (probability distributio fordiscretevariables) isasetofpairs(x i,p i )wherex i is a value of radom variable X ad p i is a probability, that a radom variable X will taes a value x i Example. Probability mass fuctio for a sigle toss of coi. Evet correspodig to heads is attributed x ; tails meas x 0. x p( X ) p( x ) x 0 p( X 0) p( x) Itroductio to probability ad statistics, Lecture 6
7 Distributio of radom variable Example. cot. Probability mass fuctio for a sigle toss of coi is give by a set of the followig pairs: { (, ), (0, )} p(x),0 0,9 0,8 0,7 0,6 0,5 0, 0,3 0, 0, 0,0 prawdopodob. zdarzeia 0,0 0,5,0 Radom variable whe discrete etails probability distributio also discrete. X Itroductio to probability ad statistics, Lecture 7
8 Probability desity fuctio Probability fuctio is itroduced for cotiuous variables; it is related to probability i the followig way: f ( x) dx P( x X < x + dx) Properties of probability desity fuctio:. f ( x) 0 +. f ( x) is ormalized f ( x) dx 3. f(x) has a measure of /x Itroductio to probability ad statistics, Lecture 8
9 Probability desity fuctio Directly from a defiitio of probability desity fuctio f(x) we get a formula of calculatig the probability that the radom variable will assume a value withi a iterval of [a,b]: P ( a < X < b) f ( x) dx b a Questio: what is a probability of xa is icorrect!!! Itroductio to probability ad statistics, Lecture 9
10 Probability desity fuctio Example. Let the cotiuous radom variable X deote the curret measured i a thi copper wire i ma. Assume that the rage of X is [0, 0 ma], ad assume that the probability desity fuctio of X is f(x)0,05 for 0 x 0. What is the probability that a curret measured is less tha 0 ma. 0,0 0,08 gestosc prawdop. 0,06 f(x) 0,0 0 f ( x) dx 0,05 P( 0 X < 0) dx Itroductio to probability ad statistics, Lecture 0 0,5 0,0 0, X
11 Quatitative descriptio of radom variables Cumulative distributio fuctio (CDF) F(x) is a probability of a evet that the radom variable X will assume a value smaller tha or equal to x (at most x) Example. cot. CDF of coi toss: F( x) P( X x) F( x 0) P( X 0) F( x ) P( X ) Itroductio to probability ad statistics, Lecture
12 Properties of CDF. 0 F ( x). F ( ) 3. F ( + ) 0. x y F ( x) F ( y) 5. F(x) has o uit 6. f ( x) df ( x) dx o-decreasig fuctio Relatioship betwee cumulative distributio fuctio ad probability desity (for cotiuous variable) Itroductio to probability ad statistics, Lecture
13 CDF of discrete variable F ( x) P( X x) f ( x ) x x f (x i ) probability mass fuctio Example.3 Determie probability mass fuctio of X from the followig cumulative distributio fuctio F(x) F ( x) 0 dla x < 0, dla x < 0 0,7 dla 0 x < dla x From the plot, the oly poits to receive f(x) 0 are -, 0,. f ( ) 0, 0 0, f ( 0) 0,7 0, 0, 5 f ( ),0 0,7 0, 3 Itroductio to probability ad statistics, Lecture 3 i i
14 CDF for cotiuous variable t F ( t) P( X t) f ( x) dx Cumulative distributio fuctio F(t) of cotiuous variable is a odecreasig cotiuous fuctio ad ca be calculated as a area uder desity probability fuctio f(x) over a iterval from - to t. Itroductio to probability ad statistics, Lecture
15 Numerical descriptors Parameters of Positio Quatile (e.g. media, quartile) Mode Variace (stadard deviatio) Rage Dispersio Expected value (average) Itroductio to probability ad statistics, Lecture 5
16 Numerical descriptors Quatile x q represets a value of radom variable for which the cumulative distributio fuctio taes a value of q. F( x ) P( X x ) f ( u) du q q Media i.e. x 0.5 is the most frequetly used quatile. x q q I example. curret I0 ma is a media of distributio. Example. For a discrete distributio : 9,,,,,, 3, 5, 6, 7 media is (middle value or arithmetic average of two middle values) Itroductio to probability ad statistics, Lecture 6
17 Numerical descriptors Mode represets the most frequetly occurrig value of radom variable (x at which probability distributio attais a maximum) Uimodal distributio has oe mode (multimodal distributios more tha oe mode) I example.: x 9,,,,,, 3, 5, 6, 7 mode equals to (which appears 3 times, i.e., the most frequetly) Itroductio to probability ad statistics, Lecture 7
18 Arithmetic average: Average value x i - belogs to a set of elemets x i x i I example.: x i 9,,,,,, 3, 5, 6, 7, the arithmetic average is,7. Itroductio to probability ad statistics, Lecture 8
19 Arithmetic average May elemets havig the same value, we divide the set ito classes cotaiig idetical elemets Example.5 x f 0, 0,0357,3 0,9, 0,07 3, 8 0,857 6, 0,9 7,5 3 0,07 9,3 0,0357, 0,07, 0,07 5, 0,0357 Sum 8 x x f + x f 0, 0,0 +,3 0, + + 5, 0,0 x 5, x Itroductio to probability ad statistics, Lecture 9 x p x p where: f, p umber of classes ( p ) Normalizatio coditio f x f
20 Momets of distributio fuctios Momet of the order with respect to x 0 m ( x ) ( xi x0) p( x 0 i i ) for discrete variables m ( x 0) ( x x0) f ( x) dx for cotiuous variables The most importat are the momets calculated with respect to x 0 0 (m ) ad X 0 m the first momet (m is called the expected value) these are cetral momets µ. Itroductio to probability ad statistics, Lecture 0
21 Expected value Symbols: m, E(X), µ,, x xˆ E( X ) i x i p i for discrete variables E( X ) x f ( x) dx for cotiuous variables Itroductio to probability ad statistics, Lecture
22 Properties of E(X) E(X) is a liear operator, i.e:. I a cosequece: E( Ci X i ) CiE( X i ) E(C) C E(CX) CE(X) E(X +X )E(X )+E(X ). For idepedet variables X, X, X Variables are idepedet whe: i E( X i ) E( X i ) i i i f ( X, X,..., X ) f( X) f( X )... ( X f ) Itroductio to probability ad statistics, Lecture
23 Properties of E(X) 3. For a fuctio of X; Y Y(X) the expected value E(Y) ca be foud o the basis of distributio of variable X without ecessity of looig for distributio of f(y) E ( Y ) y( xi ) p for discrete variables i i E( Y ) y( x) f ( x) dx for cotiuous variables Ay momet m (x 0 ) ca be treated as a expected value of a fuctio Y(X)(X-x 0 ) m ( x0) ( x x0) f ( x) dx E(( x x0) ) Itroductio to probability ad statistics, Lecture 3
24 Variace VARIANCE (dispersio) symbols: σ (X), var(x), V(X), D(X). Stadard deviatio σ(x) σ ( X ) i pi ( x E( X i )) for discrete variables σ ( X ) f ( x)( x E( X ) dx for cotiuous variables Variace (or the stadard deviatio) is a measure of scatter of radom variables aroud the expected value. σ ( X ) E( X ) E ( X ) Itroductio to probability ad statistics, Lecture
25 Properties of σ (X) Variace ca be calculated usig expected values oly:. σ ( X ) E( X ) E ( X ) I a cosequece we get: σ (C) 0 σ (CX) C σ (X) σ (C X+C ) C σ (X). For idepedet variables X, X, X σ ( C X ) C σ i ( X i i i i ) Itroductio to probability ad statistics, Lecture 5
26 UNIFORM DISTRIBUTION a x b
27 Czebyszew iequality Iterpretatio of variace results from Czebyszew theorem: Theorem: P a ( X E( X ) a. σ ( X )) Probability of the radom variable X to be shifted from the expected value E(X) by a-times stadard deviatio is smaller or equal to /a This theorem is valid for all distributios that have a variace ad the expected value. Number a is ay positive real value. Itroductio to probability ad statistics, Lecture 7
28 Variace as a measure of data scatter Big scatter of data Smaller scatter of data Itroductio to probability ad statistics, Lecture 8
29 Rage as a measure of scatter RANGE x max -x mi Itroductio to probability ad statistics, Lecture 9
30 Practical ways of calculatig variace Variace of -elemet sample: s i i x average ( x x) Variace of N-elemet populatio : σ μ N N i exp ected ( x μ) i value Itroductio to probability ad statistics, Lecture 30
31 Practical ways of calculatig stadard deviatio Stadard deviatio of sample (or: stadard ucertaity): s ( x ) i x i Stadard deviatio (populatio): σ N N i ( x ) i μ Itroductio to probability ad statistics, Lecture 3
32 Examples of probability distributios discrete variables Two-poit distributio (zero-oe), e.g. coi toss, head failure x0, tail success x, p probability of success, its distributio: Biomial (Beroulli) x i 0 p i -p p p p ( p), 0,, K, where 0<p<; X{0,,, } umber of successes whe -times sampled with replacemet For two-poit distributio Itroductio to probability ad statistics, Lecture 3
33 Biomial distributio - assumptios Radom experimet cosists of Beroulli trials :. Each trial is idepedet of others.. Each trial ca have oly two results: success ad failure (biary!). 3. Probability of success p is costat. Probability p of a evet that radom variable X will be equal to the umber of -successes at trials. p p ( p), 0,, K, Itroductio to probability ad statistics, Lecture 33
34 Pascal s triagle Itroductio to probability ad statistics, Lecture ! )! (! Symbol b a b a + 0 ) ( Newto s biomial
35 Pascal s triagle Itroductio to probability ad statistics, Lecture 35
36 Beroulli distributio Example.6 Probability that i a compay the daily use of water will ot exceed a certai level is p3/. We moitor a use of water for 6 days. Calculate a probability the daily use of water will ot exceed the set-up limit i 0,,,, 6 cosecutive days, respectively. Data: p 3 q N 6 0,, K,6 Itroductio to probability ad statistics, Lecture 36
37 Itroductio to probability ad statistics, Lecture ) 6 ( ) ( ) ( ) ( ) ( 3 6 ) ( ) 0 ( 0 P P P P P P P Beroulli distributio
38 Itroductio to probability ad statistics, Lecture (0) (6) (0) (5) (0) () 0.3 (0) (3) (0) () 0.00 (0) () (0) P P P P P P P P P P P P P Beroulli distributio
39 Beroulli distributio 0, 0,35 0,356 0,3 0,97 0,5 P() 0, 0,5 0,3 0,78 0, 0,05 0 0,033 0,000 0, Maximum for 5 Itroductio to probability ad statistics, Lecture 39
40 Beroulli distributio Itroductio to probability ad statistics, Lecture 0
41 Beroulli distributio Expected value E ( X ) μ p Variace V ( X ) σ p( p) Itroductio to probability ad statistics, Lecture
42 Errors i trasmissio Example.7 Digital chael of iformatio trasfer is proe to errors i sigle bits. Assume that the probability of sigle bit error is p0, Cosecutive errors i trasmissios are idepedet. Let X deote the radom variable, of values equal to the umber of bits i error, i a sequece of bits. E - bit error, O - o error OEOE correspods to X; for EEOO matter) - X (order does ot Itroductio to probability ad statistics, Lecture
43 Errors i trasmissio Example.7 cd For X we get the followig results: {EEOO, EOEO, EOOE, OEEO, OEOE, OOEE} What is a probability of P(X), i.e., two bits will be set with error? Evets are idepedet, thus P(EEOO)P(E)P(E)P(O)P(O)(0,) (0,9) 0,008 Evets are mutually exhaustive ad have the same probability, hece P(X)6 P(EEOO) 6 (0,) (0,9) 6 (0,008)0.086 Itroductio to probability ad statistics, Lecture 3
44 Errors i trasmissio Example.7 cotiued! ()!! 6 Therefore, P(X)6 (0,) (0,9) is give by Beroulli distributio P( X x) x P(X 0) 0,656 P(X ) 0,96 P(X ) 0,086 P(X 3) 0,0036 P(X ) 0,000 p x ( p) x, x 0,,,3,, p 0, Itroductio to probability ad statistics, Lecture
45 Poisso s distributio Itroductio to probability ad statistics, Lecture 5 We itroduce a parameter λp (E(X) λ) x x x x x p p x x X P λ λ ) ( ) ( Let us assume that icreases while p decreases, but λp remais costat. Beroulli distributio chages to Poisso s distributio.! ) ( lim lim x e x x X P x x x λ λ λ λ
46 Poisso s distributio It is oe of the rare cases where, expected value equals to variace: E ( X ) p λ Why? V ( X ) σ lim ( p p ) p λ, p 0 Itroductio to probability ad statistics, Lecture 6
47 Poisso s distributio p(x) 0, 0,35 0,3 0,5 0, 0,5 0, 0,05 0 lambda lambda5 lambda x Beroulli 50; p0,0 0,36 0,37 0,86 0,06 0,0 0,003 0,000 Poisso: λ 0,368 0,368 0,8 0,06 0,05 0,003 0,00 (x- iteger, ifiite; x 0) For big Beroulli distributio resembles Poisso s distributio Itroductio to probability ad statistics, Lecture 7
48 Normal distributio (Gaussia) Limitig case (ormal distributio) The most widely used model for the distributio of radom variable is a ormal distributio. Cetral limit theorem formulated i 733 by De Moivre Wheever a radom experimet is replicated, the radom variable that equals the average (or total) result over the replicas teds to have a ormal distributio as the umber of replicas becomes large. Itroductio to probability ad statistics, Lecture 8
49 Normal distributio (Gaussia) A radom variable X with probability desity fuctio f(x): ( x μ f ( x) exp, where - < x σ π σ < + is a ormal radom variable with two parameters: < μ < +, σ > We ca show that E(X)μ ad V(X)σ Notatio N(μ,σ) is used to deote this distributio Itroductio to probability ad statistics, Lecture 9
50 Normal distributio (Gaussia) Expected value, maximum of desity probability (mode) ad media overlap (xμ). Symmetric curve (Gaussia curve is bell shaped). Variace is a measure of the width of distributio. At x+σ ad x- σ there are the iflectio poits of N(0, σ). Itroductio to probability ad statistics, Lecture 50
51 Normal distributio (Gaussia) Is used i experimetal physics ad describes distributio of radom errors. Stadard deviatio σ is a measure of radom ucertaity. Measuremets with larger σ correspod to bigger scatter of data aroud the average value ad thus have less precisio. Itroductio to probability ad statistics, Lecture 5
52 Stadard ormal distributio A ormal radom variable Z with probability desity N(z): N( z) z exp, where - < z π < + is called a stadard ormal radom variable N(0,) E( Z) 0, V ( Z) Defiitio of stadard ormal variable Z X μ σ Itroductio to probability ad statistics, Lecture 5
53 Stadard ormal distributio Sigificace level Cofidece level Advatages of stadardizatio: Tables of values of probability desity ad CDF ca be costructed for N(0,). A ew variable of the N(µ,σ) distributio ca be created by a simple trasformatio X σ*z+µ By stadardizatio we shift all origial radom variables to the regio close to zero ad we rescale the x-axis. The uit chages to stadard deviatio. Therefore, we ca compare differet distributio. Itroductio to probability ad statistics, Lecture 53
54 Calculatios of probability (Gaussia distributio) Φ(x) 68.% pow. x (-σ, + σ) (-σ, + σ) (-3σ, + 3σ) P(μ-σ <X< μ+σ) 0,687 (about /3 of results) P(μ-σ <X< μ+σ) 0,955 P(μ-σ <X< μ+σ) 0,9973 (almost all) Itroductio to probability ad statistics, Lecture 5
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