ms2004 079-091 - Section 3.4 THEOREM 1. Consider a sequence...

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Unformatted text preview: Section 3.4 THEOREM 1. Consider a sequence of Binomial distributions, so that the nth distribution has probability p n of a success,and we assume that n → ∞ , p n → 0 and λ n = n pn → λ , t ha t for some ( )p λ >0.Then we have n x x n q n− x n ⎯n→∞ → e ⎯ ⎯ −λ λx x! for each fixed x=0,1,2,... REMARK6 If n is large the probabilities ( )p n x x n q n− x are not easily calculated. Then n λx ), provided p is small, where we replace λ be x! np. Usually poisson distribution is a good approximation of binomial distribution we can approximate them by e − λ ( when np ≤ F. THEOREM 2 Let m, n → ∞ and suppose that m/(m+n)=P m,n → P,0< P <1. Then ( )( ) → ( ) p () m x n r−x m+ n r r x x q r − x , x=0,1,2...r. 3.4.2,where B is a subset of Section 4.1 The distribution of a k-dimensional r. vector X has been defined through the relationship: PX ( B) = P( X ∈ B) , where B is a subset of R k .In particular, one may choose B to be an “interval” in R k ; i.e., B={y ∈ R k ;y ≤ x} in the sense that, if r r x = ( x1 , L , xk )' and y = ( y1 , L , yk )' , then y j ≤ x j , j = 1,….,k. We consider the case k = 1 only. PX (B) is denoted by FX ( x) (use F ( x) for brevity) and is called the cumulative distribution function of x THEOREM 1 The distribution function F of a random variable X satisfies the following properties i) 0 ≤ F ( x) ≤ 1, x ∈ R. ii) F is nondecreasing. iii) F is continuous from the right. iv) F ( x) → 0 as x → - ∞ , F(x) → 1, as x → + ∞ . REMARK 1 i) P(a <X ≤ b)=F(b )- F(a). ii) F(x-) = lim F(x n ) with x n ↑ x. iii) F(x)=F(x-) and hence P(X=x)=0 for all x. If X is discrete , its n→∞ P(X=a)=F(a)-F(a-) if F is continuous then F(X) = ∑ f ( X j ) and F(x j ) = F(x j ) - F(x j −1 ) x j ≤x iv) If X is of the continuous type, its d.f. F is continuous. Furthermore, dF ( x) = f(x) at continuity points of f, as is well known from calculus. dx Through the relations F(x) = ∫ x −∞ f (t )dt and dF ( x) = f ( x) dx We see that if f is continuous , f determines F (f=>F) and F determines f (F=>f.) Let X be a k-dimensional r. vector defined on the sample space S , and let g be a (well-behaving) function defined on R k and taking values in R m . Then g(X) is defined on the underlying sample space S, takes values in R m , and is an r. vector. Let X and X j , j=1,…..,k be functions defined on the sample space S and taking values r r in R k and R, respectively and let X = (X 1 ..X k ).Then X is an r. vector if and only if X j ,J=1,….,k are r.v.’s. THEOREM 2 Let X be an N( µ , σ 2 )-distributed r.y. and set Y= X −µ σ .Then Y is an r.v. and its distribution is N(0,1). THEOREM 3 i) Let X be an N(0,1)-distributed r.v. Then Y=X 2 is distributed as χ12 . ii) If X is a N( µ , σ 2 )-distributed r.v. ( X −µ σ ) 2 is distributed as χ12 . 4.1.1 4.1.3 4.1.5 4.1.8 4.1.14 ...
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This note was uploaded on 03/17/2010 for the course STATISTIC 472 taught by Professor Amjad during the Spring '08 term at Yarmouk University.

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