& P(A) Set I erofout com likely equally events Theory Sample A An cardonality -> subset event Set subset a Union Set Set exclusive Set complement A Set A -> partition Alor AC every frequency => P(A) Relative => element = A SE , En , Es Cermutation ↑Pr P(A) + P(B) = , Enly and > ~Pr number is a S event elements) of Joint -> Probability d = occuring not appears exactly in number of A times subset the of one occurred - number P(AUB) the UB An or (number of space An ⑲ Intersect Mutually sample ↳ comes out AcB or of possible space sample A cB =) all list the of B is that set a is space 181 I for Only - > oute - partition of times experiment was repeated -(an) of S 1 then = P(s) = P(E ) , P(E2) + Combination permutations of of ~ from n Corder is important (2 1) +(1 , , + P(E )+ , 2) = ! (2) or "Cr ~C -> number : r) ! of combinations of from (order does not matter) + ... 4(En) Probability Conditional P(F(E) It) : 4(FnE)= PCFIEP(E) PCEn)= PCE)PCE) P(FIE) P(I) 4 Law of 4 LE 4(I) = Total , En ... of I PCEn) = given P(EIP(E) : event the then independent is want] [goint Probability Enl is sample partition a P(FnEi) = probability -> E = P(+ , space s (Ei) For . any went I P(E ) : ↓ P(Ee) <CED Es P(I) PGIES) ) P(FIE) . : px(x) mass funcion P(X I = = x Ls possible value of px(x) X & Ru CDF P(E (RU) Variables -Probability PMF (pX -> = I I I Discrete Es P3 [FIEz P(FIE) Random P(Es) Ez commulative of RU Fx(x) distribution X = function discrete for P(X x) = or = continious Em4X(a) <, 0 PMF = 1 + P(E) P(+1E) + 4(Es) P (FIEs) Quantile the quantile 2 Independent RV value X of such Ex (x2) that RVs X Y and P(X x , Y , y) = = P(X x) = = - 4(X = RU of P(t · = - ) 1) [ECx (Mx) - Expectation of marginal -> marginal ~Joint = independent are 4(X x(+ y) Discrete smallest the is Mean -> value of X : < xP(X E(x: E(g(x)) Variance ox Standard a x) < g(x)px(x) RV (02) = of = EG(X-nx)"y = 6 -(ax + by E(ax bi . Combinations linear of ↑Y ~x = - Expectations - (ox) Deviction ↑ , E(x2) = + : c) ab = . sE(x) E(x) · RVs of E(y) + bE(y) -> + c only for Independent RVs >, < Variance (ax Var by + + ) broi-za(x act = RVs Independent for ) : Distribution Bernoulli -fail [0 17 Rx Combination Coneur of random discrete -> , pass or X variable Bernoulli I distributed p Mx RU for 6x" RU for Binomial for X x RV fow RU for CDT = * MF x P(X = < = Rx rate My oh 2 0, 1 parameter for for , 2, Bin(n - qx(x) = P - E(x = p - p2 = q(1 a) - b) within distribution (2)p"(1 p) independent trials and ip' probability of success in each trial the - x - = - c) x) Bon (n ... (x) RUX- , q) (n , -> p) binocdt(c = binopdf = up up(1 a) -> , (c - n , , n p) , c) (X-40(x) Poisson Distribution = E(x2) - > - = this , X-Bin P (x Mx parameter stration ↳ G0 17 = xx(x) o (x binomial ~ -> (p) Bemoulli Rx Mx (p) Bernoulli ~ Distribution RU a X (p) as Y -> po(i) RUX-po(i) Px(x) ↑ >0 -> - Mx Gr = P(X x) = = CDT i : = x * MF = = P (x P(X < = c) x) = = i) (c i) poisscdt (c poisspdf , , MA , number of success wanted ->') Multivariate * Px (Xi = y(x g) y S x L a + I btt c+ Px(0) n 4xC ↓+h g Spi xn , a + = b + c + (Here , = , xs , =x with >2 multinomial (n ; distribution xn)" .... are - Easi Xe , = an , ..., Xxx = x) = , strength the =[x4) +) outcome possible p , p -, ..., - = E(x) cy · cap !***...** of multivariate ECt) 4(X = x = (1) , += y) " o as perfect ) . (like throwing n = dependence of . ai) Correlation and E(x1 car(x conditional dependent) events the I measures Cor(x Y -> d ... This -g) = ctdteff : = = p(X Lovariance (ly) I bonomal of (X = Px1y d Distribution Generalization PMF joint - Py(a) Multinomial = y) C Py(s) * = 9 7 - e y , b e Py x 6 9 D = , Xn) . . , P(X = , , RV >perfect poite correlation RUs a dice : six outcomes Continuous PDF - PMF -- CDA F(x F, - F(x) falt) a(x-x) = 1 it fact de % (x = (time = = inc 1 - - to x)" an fx(x) event - dxa excel = E(x2) - E(X) (i) ** a(X-f xxa = fies- (ii = eine Y 2 = Memoryless Property Uniform #(c) = Yx = x (A) = Distribution in >0 war + (x) E(x-mx)" = Exponential M fx(x)s0fff(x)dt ceace" x = var(X) Variable Random of Distribution = ExpRV 4(Xxx+a(X <a) describes a State . RV with ]" - Rx= = 4(x> x) [a b] , ~ 4[a b] , Mx = 62 = ( Standard H Normal 6 0 = x X N - Co Symmetry NC0 1) Rule , NCM - P(X-M) Multivariate of there (t)" = for , NC0 1) , fake) : M ! ! 4(Xxx) - 62) : X RU 4 1 , distribution , = N(n ~ 1 fx(x) of = RUX , 1) , 4(X -) 2 G = Distribution o , , 4) = RV a 0 95 = . , Distribution Normal are ) 2 hence N X wenite X x = = (M , (X - Xm Mxn , , Xa) ... , ,. (Mx deviation are used as bars error 2) N(x 2) ~ standard I2 ..., , 2) y = (y -> covariance I cou(x I = , matrics x) . 0 x cov , y Y AX + , b < is -, x1) culx , xn) the mx matrice (x : ... < mxn cov ... diagonal The = x X of I =x rector My 2 = = y (x , x2 , x)" Mx = (1 , 0 .. 7" A A E(X + 1= ) = x = = (2) +1 = X, - xn + x- ( =:)( : :) Y + = X b cor(X) AT = AAE X , + 0 - x3 (i) + y 2 = (i) ( : = :)( : =i)(i :7 (i) = X of variance , Max - I = 5 X-Ni (4 or X eg " = ( = ) (i) (i) : : : Centeral IID RUs X E(Xi) (xi) X , x = , ..., + (X()) Central the all for function RUs (2) var() = = = I: X .. follows - that w(n i) , ~N(n : , now normal distributed variable Data Collection Data Handling Inference gr : si 2:= are . S Er= ID the Since var(tx) +ID are m = +M- You (ai+st - -> x() -> they since distributed e An : : lamot theorem Statics X . of RVs the all identically and R deviation/enor Standard "a M for independent - Xn 2 = I(x(r)) war Incorem mean X(u) = Samle var - Limit . c)")-ni (gie e Var(X(a)) ↓ as n4 st = = = Estimator (P) RUP p estimate Unbiased (estimator) in = observed Estimator Ex Confidence If into = 1(rx) was C1 the M = Interval experiment an c = M - (P) Estimate is = m and done 95 % the of p (2) relevant if , 17 2 = norminCitiea X (i = -t volume . 19 = -> (0x )) NCM , 6) estimate with : p 1 - C quantile= e ⑨ launts N c1 0 = a times quantile NCo level confidence with · I I relevant Carea Ch 1- were quantile quantile) relevant I = 0 . 17 and CI (0 , 14 , % 0 2 . and c = 0 . 95 this means that of the experiment is repeated " # of times p will fall least square estimator Y = -Zili E N it ar best Nut