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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 kdimensional 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 kdimensional r. vector defined on the sample space S , and let g be a
(wellbehaving) 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.
 Spring '08
 AMJAD

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