Theory of Probability and Statistics II Explained

STATISTICS-II by Dr. S. SEYADALI FATHIMA Assistant Professor Department of Mathematics Statistics II III B.Sc Mathematics III B.Sc Mathematics

THEORY OF PROBABILITY Random Experiment: The random experiment is an experiment inwhich The set of all possible outcomes are known. But exact outcome is not known. Ex: Ex:- -Tossing a coin, Rolling a die. Tossing a coin, Rolling a die.

Sample Space: The set of all possible outcomes in a random experiment is called sample space. It is denoted by (S). Ex:- When tossing a coin S={H,T} When rolling a die S={1,2,3,4,5,6} Sample point: Each element of a sample space is called Sample Point.

Let {Ai} be a sequence of mutually exclusive and exhaustive events in a sample space S such that P(Ai)>0 for all i. Let B be any event with P(B)>0. Then P(Ai/B)= [P(Ai) P(B/Ai)]/[ P(Ai) P(B/Ai)]. Proof:- By Baye s formula we have, P(Ai/B)=[P(Ai) P(B/Ai)]/[P(B)] -------(1) We claim that P(B)= P(Ai) P(B/Ai) Since the events Ai are mutually exclusive and exhaustive we have, UAi=S and Ai s are disjoint.

Therefore, B=B S=B (UAi) =U(B Ai) P(B)=P[U(B Ai)] = P(B Ai) P(B)= P(Ai)P(B/ Ai) [since P(B/A)=[P(B A)]/[P(A)]] Sub P(B) in equation (1) we get, P(Ai/B)=[P(Ai)P(B/Ai)]/[ P(Ai)P(B/Ai)]

ADDITION LAW IF A AND B ANY TWO EVENTS OF A SAMPLE SPACES. THEN P(AUB)=P(A)+P(B)-P(A B) AUB=AU(A B) P(AUB)=P(A)+P(A B)-P(A A B)= P(AUB)=P(A)+P(A B) ---------(1) Now, B=(A B)U(A B) A B and A B are disjoint events. P(B)=P(A B)+P(A B) P(A B)=P(B)-P(A B) ------------(2) (2) in (1) we get, P(AUB)=P(A)+P(B)-P(A B)

Let A1,A2,An be n events in a Sample Space S. Then P(A1 A2 A3 .. An) =P(A1)P(A2/A1)P(A3/(A1 A2) .P(An/A1 A2 A3 An-1) P(A1)P(A2/A1) P(An/A1 A2 An-1) =P(A1).P(A1 A2)/P(A1) ...P(A1 A2 . An)/ P(A1 A2 . An-1) =P(A1 A2 .. An)

If A and B are any two events in a Sample Space. Then prove that P(A B) 1-P(A)-P(B) ii. Generalized Boole s Inequality. For the events A1,A2, ..,An, . in a Sample Space. Then P( Ai) 1- P(Ai) i. 1) For any two events A and B we have P(AUB)=P(A)+P(B)-P(A B) P(A B)=P(A)+P(B)-P(AUB) =(1-P(A))+(1-P(B))-P(AUB)

=1-P(A)-P(B)+[1-P(AB)] [since 0 P(AUB) 1] 1-P(A)-P(B) 2) Let B1= Ai so that Ai=A1 B1 Now, P( Ai)=P(A1 B1) 1-P(A1)-P(B1) [By (1)] =1-P(A1)-P( Ai) =1-P(A1)-P(UAi) 1-P(A1)-P(Ai) 1- P(Ai)

F(x)=P(Xx)=f(xi)=P(X=xi) If X takes only a finite number of values {x1,x2, ,xn} where (x1<x2< ..<xn) Then, 0 - x x1 f(x1) x1 x<x2 F(x)= f(x1)+f(x2) x2 x<x3 .. f(x1)+f(x2)+ +f(xn) xn x<

A random variable X is said to be continuous random variable if it can take any value in an interval which may be finite or infinite. Let X be a continuous random variable taking the values in the interval (- , ). Let f(x) be a function such that f(x) is integral (- , ) f(x) 0 for all x (- , ) f(x) dx=1 Then f(x) is called the p.d.f of the Continuous random variable X.

PROBLEM A random variable X has the following probability density function Xi -2 -1 0 1 2 3 p(xi) 0.1 k 0.2 2k 0.3 k Find the value of k. I. Mean II. Variance III. P(x>=2) IV. P(x<2) V. P(-1<x<3)

SOLUTION: pi =1 0.1+k+0.2+2k+0.3+k=1 0.6+4k=1 4k=1-0.6 K=0.1 2). Mean: E(X)= xi pi=(-2)(0.1)+(- 1)(0.1) +(0)(0.2)+(1)(0.2)+(2)(0.3)+(3)(0.1)

=-0.2-0.1+0+0.2+0.6+0.3 =-0.3+0.11 E(x)=0.8 3). Variance E(x2)=[(4)(0.1)+(1)(0.1)+(1)(0.2)+(4)(0.3)+(9)0.1)] =0.4+0.1+0+0.2+1.2+0.9 =2.8

2=E(x2)-(E(x))2 =2.8-((0.8))2 =2.8-0.64 =2.16 4). P(X>=2)= P(X=2)+P(X=3) =0.3+0.1 =0.4 5). ( P(X<2)= P(X=1)+P(X=0)+P(X=-1)+P(X=-2)

=0.2+0.2+0.1+0.1 =0.6 6). P(-1<X<3)=P(X=0)+P(X=1)+P(X=2) =0.2+0.2+0.3 =0.7

Let X be Discrete random variable which can assume any of the values x1,x2, .,xn, with corresponding probabilities. Pi=P(X=xi)= Pi=P(X=xi)=i i=1,2,3, The, the mathematical expectations of X. Denoted by E(x)= Pi xi E(x)= Pi xi provided the series is absolutely convergent. =1,2,3,

If X and Y are random variables. Then, 1) E(C)=C, where C is constant. 2) E(CX)=CE(X), where C is constant. 3) E(Ax +b)=a E(X)+b 4) E(X+Y)=E(X)+E(Y) 5) E(XY)=E(X)E(Y) if X and Y are independent random variable. 6) Let X be a discrete random variable having a p.d.f Pi. Let (x) be a function of X. Then E( (x))= Pi (xi)

CUMULANT GENERATING FUNCTION (C.G.F) DEFINITION The cumulant generating function Kx(t) of a random variable X is defined by Kx(t)=loge (Mx(t) provided the right hand side can be expanded as a convergent series in powers of t. Hence Kx(t)=loge Mx(t)=k1t+k2t2/2!+ ..+kr tr/r!+ And kr=coefficient of tr/r! in Kx(t) is called the rth cumulant of X.

BINOMIAL DISTRIBUTION Let n be any positive integer and let 0<p<1. Let q=1-p. Define p(x)= ncx px qn-x if x=0,1,2, ..n 0 Otherwise A discrete random variable with the above p.d.f is said to have binomial distribution and the p.d.f itself is called a binomial distribution.

NOTE NOTE 1 1 The two independent constants n and p in the distribution are known as the parameters of the distribution. If X X is a binomial variate with parameters n and p we write as X~B(n , p). NOTE 2 NOTE 2 The probability distribution function of a binomial distribution is obtained by considering n independent trials of a random experiment whose outcome is success or failure.

ADDITION PROPERTY OF BINOMIAL DISTRIBUTION If x1~B(n1,p) , x2~B(n2,p) are independent random variables If x1~B(n1,p) , x2~B(n2,p) are independent random variables then x1+x2 is B(n1+n2,p) then x1+x2 is B(n1+n2,p) Proof: Proof: Given x1 and x2 are independent random variables with parameters. N1,p and n2,p respectively. Let consider m.g.f of x1 and x2 about origin. Mx1(t)=(q+pet)n1 and Mx2(t)=(q+pet)n2.

Now, Mx1+Mx2(t)=Mx1(t)-Mx2(t) = [since x1 and x2 are independent] =(q+pet)n1(q+pet)n2 =(q+pet)n1+n2 =m.g.f of the binomial x1+x2 with parameters n1+n2 and p. Hence by uniqueness theorem x1+x2 is a binomial variable with parameters n1+n2 and p.

MODE OF BINOMIAL DISTRIBUTION CASE 1: CASE 1: If (n+1)p not an integer clearly mode is the integer part of (n+1)p i.e.). [(n+1)p] is the mode and distribution is unimodel. CASE 2: CASE 2: If (n+1)p is an integer both (n+1)p and (n+1)p-1 will represent mode and the distribution is bimodel.

REFERENCE BOOK: STATISTICS II By Dr.S.Arumugamand Mr. A. Thangapandi issac REFERENCE BOOK: STATISTICS