1. Bibliography Agratini, O., Blaga, P., Coman, Gh., Lectures on Wavelets, Numerical Methods and Statistics, Casa Crtii de Stiint, a a Cluj-Napoca, 2005. Blaga, P., Calculul probabilitilor i statistic matematic. at s aa Vol. II. Curs i culegere de probleme, Universitatea s Babe-Bolyai, Cluj-Napoca, 1994.s Blaga, P., Statistic. . . prin Matlab, Presa Universitar aa Clujean, Cluj-Napoca, 2002.a Blaga, P., Rdulescu, M., Calculul probabilitilor, aat Universitatea Babe-Bolyai, Cluj-Napoca, 2002.s Lisei, H., Probability theory, Casa Crtii de Stiint,a a Cluj-Napoca, 2004. Lisei, H., Micula, S., Soos, A., Probability Theory trough Problems and Applications, Cluj University Press, 2006. Shiryaev, A. N., Probability (2nd ed.), Springer, New York 1995.

2. Probability spaceE (Experiment) (Outcomes - results of the experiment) is called sample spaceDenitionA non-empty subset K P () is a -algebra (-eld) if itsatises the following conditions: 1o if A K, then A K (contrary event), 2o if Ai K, i I , then the union Ai K,iIand the pair (, K) is called the space of events (measurable spaceof events). 3. Denition Let (, K) be a measurable space of events. The set functionP : K R, is called probability, if it satises the conditions:1o P (A) 0, A K,2o P () = 1,is the certain event,3o P is additive, i.e. for any subset of events Ai K, i I , pairwise disjoint events (mutually exclusive events), Ai Aj = (impossible event), i = j, it is satised P Ai = P (Ai ) ,iIiIand the triple (, K, P) is called probability space. 4. Proposition (1) P () = 0, (2) P A = 1 P (A),A) = P (B) P (A B), (3) P (BA) = P (B) P (A), if A B, (4) P (B P (B) , if A B, (5) P (A) (6) P (A B) = P (A) + P (B) P (A B),nn (7) PAiP (Ai ). i=1i=1 5. Property (Poincar)e nn nP (Ai ) P (Ai Aj ) + P Ai = i=1i=1 i,j=1 i 0, it is called conditional probability of the event A withrespect to the event B the ratio P (A B) PB (A) = P (A | B) = P (B) 7. Proposition Let (, K, P) be a probability space, then (1) for B K, with P (B) > 0, the function set PB : K R, dened by P (A B) PB (A) = P (A | B) = , A K,P (B)is a probability, and the triple (, K, PB ) is a probability space, (2) A, B K, satisfying the condition P (A) P (B) > 0, we haveP (A B) = P (A) P (B | A) = P (B) P (A | B) , (3) A K, satisfying the condition 0 < P (A) < 1, we have P (B) = P (A) P (B | A) + P A P B | A , B K. 8. Property (Multiplication Formula for Probabilities) Let us consider the events Ai K, i = 1, n, satisfying P (n Ai ) > 0, theni=1 n n1= P (A1 ) P (A2 | A1 ) P (A3 | A1 A2 ) . . . P An P Ai Ai . i=1i=1 9. Property (Formula for Total Probability) Let A K be an event and a complete set of disjoint events, (Ai )iI , Ai K, then P (Ai ) P (A | Ai ) .P (A) =iIProperty (Bayess Formula) Let us consider (Ai )iI , Ai K, a complete set of disjoint events, and the event A K, with P (A) > 0, then P (Ai ) P (A | Ai )P (Ai | A) =, for all i I .P (Ai ) P (A | Ai )iI 10. Independence DenitionThe events A, B K are called independent, if P (A B) = P (A) P (B) .PropertyLet (, K, P) be a probability space and the events A, B K,(1) if P (A) = 0 or P (A) = 1, then the events A and B areindependent,(2) if the events A and B are independent, thenP (A | B) = P (A), and P (B | A) = P (B),(3) if the events A and B are independent, then pairs of events A, B , A, B and A, B are independent. 11. Denition The events (Ai )i=1,n are called independent, if for all 1 i1 < i2 < < ik n, and k = 2, n, the following relations are satised P (Ai1 Ai2 Aik ) = P (Ai1 ) P (Ai2 ) . . . P (Aik ) . Denition Let (, K, P) be a probability space. The events of the subset (Ai )iI , Ai K, are called independent, if PAj = P (Aj ) , jJjJfor all nite subsets of indices J I . 12. Random VariablesLet (, K, P) a probability space.DenitionA real function X : R, is called random variable, if X 1 ((, x)) = (X < x) = X () < x K, for all x R. RemarksIf |X ()| denotes the cardinal of the range of random variableX , then the random variable is called: Discrete random variable, if |X ()| 0 , i.e. it is a countable (denumerable) set; Simple random variable, if |X ()| < 0 , i.e. it is a nite set; Continuous random variable, if |X ()| = , i.e. it is a non-numerable (uncountable) set. 13. Random VectorDenition A real valued vector functionX = (X1 , . . ., Xn ) : Rnis a random vector (ndimensional random vector), if (X < x) = (X1 < x1 , . . . , Xn < xn )= X1 () < x1 , . . . , Xn () < xn K,for all x Rn . 14. Distribution FunctionLet us consider the probability space (, K, P) and a randomvariable X : R.DenitionThe real functionF : R R,dened by x R,F (x) = P (X < x) , (1)is called the (cumulative) distribution function of X . RemarkSome books consider this denition for distribution function,others (Matlab) use the denition given byx R.F (x) = P (Xx) , 15. Proposition Let X be a random variable with the corresponding distribution function F . Then we have: (1) x R, 0 F (x) 1, (2) a, b R, a < b,X < b) = F (b) F (a), P (a P (a < X < b) = F (b) F (a) P (X = a), b) = F (b) F (a) P (X = a) + P (X = b), P (a < X b) = F (b) F (a) + P (X = b), P (a X (3) x1 , x2 R, x1 < x2 , F (x1 )F (x2 ) (F is undecreasing function), (4)lim F (x) = F () = 0, x (5)lim F (x) = F (+) = 1, x+ (6) x R,F (y ) = F (x 0) = F (x) (F is left-continuous). limy x 16. Remark If F is piecewise constant, then the points of discontinuity xi , i I , are the values of a discrete random variable X , and the corresponding values of jumps are given by the probability distribution of X ,i I.pi = P (X = xi ) , 17. Denition Let X = (X1 , . . ., Xn ) a random vector. The real functionF : Rn R, dened byx Rn . F (x) = F (x1 , . . . , xn ) = P (X1 < x1 , . . . , Xn < xn ) ,is called the (cumulative) distribution function of the random vector X. Denition Let X = (X1 , . . ., Xn ) be a random vector. All distribution functions of the random vectors (Xi1 , . . ., Xik ), 1 i1 < . . . < ik n, k = 1, n 1, are called marginal distribution functions of the random vector X. Property If F is the distribution function of random vector X, then the distribution function of random vector (Xi1 , . . ., Xik ) is given by Fi1 ,...,ik (xi1 , . . ., xik ) =lim F (x1 , . . ., xn ) . xj j=i1 ,...,ik 18. Probability Density FunctionDenitionA random variable X is called (absolutely) continuous if thecorresponding distribution function F is absolutely continuous, i.e.there exists a function f : R R, such thatx for all x R. F (x) = f (t) dt, The function f is called (probability) density function.PropositionIf the random variable X is continuous, having the distributionfunction F and density function f , then: x R, f (x)(1) 0,(2)F (x) = f (x), a.e. (almost everywhere) on R, b(3)for a < b, P (aX < b) = f (x) dx, a +(4) f (x) dx = 1. 19. Remarks Let X be a continous random variable, then P (X = a) = 0, for each a R. It follows, in this case, that P (aX < b) = P (a < X b) = P (a < X < b)b= P (a Xb) =f (x) dx.aIf the continuous random variable X has the distribution function F and the density function f , then we have successivelyF (x + x) F (x) f (x) = F (x) = limxx0 P (x X < x + x) = lim xx0Therefore, for small values of x, we have X < x + x) f (x) x. P (x 20. Denition A random vector X = (X1 , . . ., Xn ) is called (absolutely) continuous if the corresponding distribution function F is absolutely continuous, i.e. there exists a function f : Rn R, called (probability) density function, such thatx1 xn F (x) = F (x1 , . . ., xn ) = ...f (t1 , . . ., tn ) dt1 . . .dtn , for all x = (x1 , . . ., xn ) Rn . Proposition Let X = (X1 , . . ., Xn ) be a continuous random vector, having the distribution function F and the density function f , then f (x) 0, for all x Rn , (1) n F (x1 , . . ., xn ) (2) = f (x1 , . . ., xn ), a.e. (almost everywhere) x1 . . . xn on Rn ,if D Rn , we have P (X D) = (3) f (x) dx, D 21. Denition Let X = (X1 , . . ., Xn ) a continuous random vector. All densities functions of the random vectors (Xi1 , . . ., Xik ), 1 i1 < . . . < ik n, k = 1, n 1, are called marginal densities functions of X. Property Let f be the density function of the continous random vector X, then the density function of random vector (Xi1 , . . ., Xik ) is given by fi1 ,...,ik (xi1 , . . ., xik ) =f (x1 , . . ., xn ) dxj1 . . .dxjnk ,Rnkwhere {j1 , . . ., jnk } = {1, . . ., n}{i1 , . . ., ik }. Remark The density function of Xi is given by xi R. fi (xi ) = f (x1 , . . ., xn ) dx1 . . .dxi1 dxi+1 . . .dxn , Rn1 22. IndependenceDenitionLet X = (X1 , . . ., Xn ) be a random vector having distributionfunction F . The random variables X1 , . . ., Xn are independent if F (x) = F (x1 , . . ., xn ) = FX1 (x1 ) . . .FXn (xn ) , for all x = (x1 , . . ., xn ) Rn , FXi being the distribution function ofrandom variable Xi . 23. Remarks If X = (X1 , . . ., Xn ) is a discrete random vector, then the random variables X1 , . . ., Xn are independent if and only if P (X1 = x1 , . . ., Xn = xn ) = P (X1 = x1 ) . . .P (Xn = xn ) ,for all xi Xi (), i = 1, n. If X = (X1 , . . ., Xn ) is a continuous random vector, then the random variables X1 , . . ., Xn are independent if and only if the density function of random vector X satises the relation f (x) = f (x1 , . . ., xn ) = fX1 (x1 ) . . .fXn (xn ) ,for all x = (x1 , . . ., xn ) Rn , fXi being the density function of random variable Xi . 24. Proposition If the continuous random vector (X , Y ) has the density function f , then the density functions for the random variables sum, product, and ratio are respectively:+f (u, x u) du,x R, fX +Y (x) = + x du x R,fXY (x) = f u, , |u| u +f (xu, u) |u| du, x R. fX /Y (x) = If the random variables X and Y are independent, then +fX (u) fY (x u) du,x R,fX +Y (x) = + x du x R,fXY (x) = fX (u) fY, |u| u +fX (xu) fY (u) |u| du, x R. fX /Y (x) = 25. Discrete Uniform Distribution The random variable X has discrete uniform distribution, when thedistribution isx 1where N N;X ,f (x) = f (x; N) = , x = 1, N.1 NN x=1,N The distribution function is0,if x 1, k F (x) = F (x; N) =, if k < x k + 1, k = 1, N 1,N1,if x > N, 26. The distribution of random variable X is given byf(x)1/N 0 1 2 3 4N1 N x f(x)=1/N, pentru x=1,...,N The graph of distribution function F isF(x) 1 (N1)/N 2/N1/N 0 1 2 3N1 N x 27. Binomial DistributionThe random variable X has binomial distribution, we denoteB (n, p), when the distribution isxn x nx X , f (x) = f (x; n, p) = pq , x = 0, n, x f (x) x=0,nand p (0, 1), q = 1 p.The random variable X represents the number of successes in nindependent trials of an experiment.Binomial distribution was descovered by James Bernoulli, and waspresented in his book Ars Conjectandi (1713).1Pascal considered the particular case p = 2 The distribution function is given by0,if x 0,k1n i niF (x) = F (x; n, p) =p q , if k 1 < x k, k = 1, n, i i=01,if x > n, 28. In Figure, the vertical bars represent the probabilities of binomial distribution B (5, 0.4) and the corresponding distribution function. f(x)F(x) 0 1 2 3 4 5 x 0 12 345 x f(x)=(n)pxqnx, pentru x=0,...,nx 29. Poisson DistributionThe random variable X has Poisson distribution, we denotePo (), when distribution is given byxx X , where f (x) = f (x; ) =e , and > 0. x!f (x) x=0,1,2,...The random variable X , having Poisson distribution, gives thenumber of occurrences of a xed event in a time interval, on adistance, on an area, and so on.Poisson (1837) proved that this distribution is a limit case ofbinomial distribution.Namely, when p = p (n) and np , for n , one obtains thatx n x nx f (x; n, p) = pqe. xx!One remarks that for high values of n, p has to be small, to holdon the product np constant (). Thus, the probability p ofconsidered event is small, when n is high.It is the reason that this distribution to be called the distribution ofrare events. 30. The Figure illustrates this remark. It was considered binomial distribution B (100, 0.1) and corresponding Poisson distribution Po (10), because np = 10.Probabilitatile distributiilor 0.14Legea binomialaLegea lui Poisson 0.120.1 0.08 0.06 0.04 0.0246810 12 14 16 18 We also remark a known result, which establishes the connection between the Poisson distribution and exponential distribution! When Poisson distribution gives the number of occurrences in a time interval, the exponential distribution gives the length of the interval between two successsive occurrences of events. 31. Uniform distributionThe random variable aleatoare X has a uniform distribution on theinterval [a, b], we denote by U (a, b), when the density function is 1 , if x [a, b], f (x) = f (x; a, b) = b a(2) if x [a, b].0, / The distribution function of X is 0,if x < a, xx a , if a x b,F (x) = F (x; a, b) =f (t; a, b) dt =b a1,if x > b. The name is in connection with the fact that if one considerssubintervals of the interval [a, b] with the same length , then theprobability that X belongs to such intervals is / (b a). 32. The Figure contains the graphs of density function and distribution function. f(x)F(x)1 1/(ba) ab xa b x We remark that distribution function is continuous. Thus, the values x = a and x = b in formula (2) can be attached to the cases F (x) = 0 and F (x) = 1, respectively. It follows that density function can be considered non-zero on [a, b] or (a, b] or [a, b) or (a, b). In all cases we have uniform distribution U (a, b). 33. Normal DistributionThe random variable X has normal distribution (Gauss-Laplacedistribution) with the parameters R and > 0, we denoteN (, ), when the density function is (x)21 f (x) = f (x; , ) = e 22 ,for all x R. 2The corresponding distribution function is given by xx(t)2 1 f (t; , ) dt = F (x) = F (x; , ) =edt, 2 2 2 for all x R. 34. The graphs of density function and distribution function are given in the Figure. The curve which represents the graph of density function f is called the curve of Gauss.f(x)F(x)fmF(+) fiF()=0.5F() + +x x1 For x = is obtained the maximum of f , fm = 2 . The inexion points of f are x = with the corresponding value f ( ) = fi = 1 2e 35. The distribution function of standard normal distribution, N (0, 1), is x12 t2 (x) = e dt, x R,2 and is called Laplace function. We remark that the function x 1 2 t2 (x) = e dt, x R, 2 0 is also called Laplace function. Between the two Laplace functions the following relation 1 (x) = + (x) , 2 holds. Using the two Laplace functions we have:xx(t)2 1 f (t; , ) dt = F (x; , ) =e 22 dt 2 x 1 x == + .2 36. 2 distribution (Helmert-Pearson) The random variable X has 2 distribution or Helmert-Pearsondistribution, and one denotes 2 (n), when the density function is 1 nxx 2 1 e 2 , if x > 0, nn 22 f (x) = f (x; n) =2 0, if x 0, where n N represents the number of degrees of freedom. 37. The Figure contains the graphs of density function of 2 (n) distribution, for some values of parameter n. f(x;n)n=5n=10n=20f(x;5) f(x;10) f(x;20) 3 818 x Property If the random variables X1 , . . . , Xn are independent, each of them having the same normal distribution with the parameters = 0 and = 1, then the random variable n2 Yn =Xk ,k=1is 2 (n) distrubuted. 38. t distribution (Student) The random variable X is t-distributed (Student) distributed,denoted by T (n), when it has the density function n+1 n+1x2 22 x R, f (x) = f (x; n) = 1+, nnn2 where n N represents the number of degrees of freedom.Gosset (1908) descovered this distribution.The result was not published at the rst time, but using thepseudonym Student, then it was published.For n = 1, it is obtained Cauchy distribution:1x R. f (x) =, (1 + x 2 ) 39. The Figure contains the graphs of density of T (n), for some values of n.f(x) n=1n=3n=205 4 3 2 10 1 2 3 45 x Property If the random varibales X and Y are independent, X is normally distributed, and Y having 2 distribution with n degrees of freedom, then the random variableXT=Y /nhas the Student distribution with n degrees of freedom. 40. F distribution (FisherSnedecor) The random variable X has F (Fisher-Snedecor) distribution,denoted by F (m, n), when the density function is m+nm m m 1 m m+n2 22x21+ x, x > 0, m nn nf (x) = f (x; m, n) = 2 20, x 0, where m, n N represent the numbers of degrees of freedom. 41. The Figure contains graphs of density function of F (m, n), for some values of the parameters m and n. f(x) m=4, n=2 m=3, n=100.71483 m=7, n=10 0.595x Property If the random variables X and Y are independent, with 2 (m) and 2 (n) distributions, then the random variable X Y F=m nis F (m, n) distributed. 42. Multidimensional normal distributionRandom vector X = (X1 , . . ., Xn ) has n-dimensional(nondegenerate) normal distribution, one denotes N (, V), whenthe density function isf (x) = f (x1 , . . ., xn ) = f (x; , V)1 1 exp (x ) V1 (x ) ,=n 12(2) 2 [det (V)] 2for all x Rn .V is a positive denite matrix of order n, and Rn .When n = 2, the density function of the random vector (X , Y ),having the two-dimensional normal distribution can be put in theform1 f (x, y ) = 21 2 1 r 2 (x 1 )2 (x 1 ) (y 2 ) (y 2 )2 1 exp 2r+, 2 2 2 (1r 2 ) 1 212for (x, y ) R2 . 43. The Figure represents the graph of density function of two-dimensional normal distribution with 1 = 2 = 0, 1 = 1, 2 = 2 i r = 0.5.s 0.07 0.06 0.05 0.04 f(x,y)0.03 0.02 0.01 06 4 3 2 2 0 1 0 214 2 6 3 y x Property If the random vector (X , Y ) is a two-dimensional normally distributed, N (1 , 2 ; 1 , 2 ; r ), then each of the components X and Y of random vector are normally distributed: N (1 , 1 ) and N (2 , 2 ) respectively. 44. Conditional distributionLet (X , Y ) be a two-dimensional random vector, having thedistribution function F .DenitionThe conditional distribution function of the random variable Xwith respect to random variable Y , is the function FX |Y : R R,given byFX |Y (x|y ) = P (X < x | Y = y ) ,x R, y R, xed.RemarkWe can rewrite FX |Y (x|y ) = lim P (X < x | y Y < y + h) ,h 0 and using the denition of conditional probability P (X < x, y Y < y + h)FX |Y (x|y ) = limP (y Y < y + h)h0F (x, y + h) F (x, y ) = limh 0 FY (y + h) FY (y ) 45. Let (X , Y ) be a discret random vector, having the distribution X Y ... yj.... .. .. .xi ...pij... . . . . . .with (xi , yj ) R R, (i, j) I J. Denition The conditional distribution of random variable X with respect to random variable Y has the distribution given byP (X = xi , Y = yj ) pij pi|j = P (X = xi | Y = yj ) =(i, j) I J,= , P (Y = yj )pjwhere j J.p j = P (Y = yj ) = pij ,iI 46. We remark that the formula for all i I ,pi = P (X = xi ) = p j pi|j , jJholds. It represents the formula for total probability. We have also the Bayess formula: pi pj|i pij(i, j) I J. pi|j == , pij pi pj|i iI iI 47. Let (X , Y ) be a two-dimensional continuous random vector, having the density function f . Denition The conditional density function of random variable X with respect the random variable Y , is the function given by f (x, y ) , if f (y ) = 0, Y fY (y )fX |Y (x | y ) = 0, if fY (y ) = 0. The formula for total probability and the Bayess formula hold: +x R,fX (x) =fY (y ) fX |Y (x|y ) dy ,fX (x) fY |X (y |x) (x, y ) R R.fX |Y (x|y ) =, fX (x) fY |X (y |x) dx R 48. Example Let (X , Y ) be a two-dimensional normally distributed random vector, N (1 , 2 ; 1 , 2 ; r ). The components X and Y are normally distributed: N (1 , 1 ) and N (2 , 2 ) respectively. The conditional densities of X with respect to Y and Y with respect to X are: (xm1 (y ))2122e 2(1r )1 , (x, y ) R R, fX |Y (x|y ) = r 2) 2 (1 1 (y m2 (x))21 2(1r 2 ) 2 fY |X (y |x) = (y , x) R R,e2, 2 (1 r 2 )2 where 1 2m1 (y ) = 1 + r (y 2 ) , m2 (x) = 2 + r (x 1 ) . 2 1 We remark that the two conditional densities correspond to the normal distribution: N m1 (y ) , 1 1 r 2 and N m2 (x) , 2 1 r 2 respectively. 49. In the Figure are presented the graphs of the two conditional densities functions, when a two-dimensional normal distribution with 1 = 2 = 0, 1 = 1, 2 = 2, r = 0.5 is considered, for three values of the two components: X = 2, 0, 2 and Y = 2, 0, 2. X = 2 Y = 2 fX|Y(x|y)X=0 Y=0 fY|X(y|x)X=2 Y=2x6y32/3 0 2/3 36 31.5 0 1.5 3