Introduction to Mathematical Statistics |
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Page 15
... Pr ( XeA ) can be written as = Pr ( a < X < b ) = ƒ f ( x ) dx . Moreover , if A = { x ; x = a } , then = = S Pr ( XeA ) Pr ( X · S f ( x ) dz = 0 , a ) = √ ƒ ( x ) P ( A ) since the integral ( f ( x ) = = f ( x ) dx is defined in ...
... Pr ( XeA ) can be written as = Pr ( a < X < b ) = ƒ f ( x ) dx . Moreover , if A = { x ; x = a } , then = = S Pr ( XeA ) Pr ( X · S f ( x ) dz = 0 , a ) = √ ƒ ( x ) P ( A ) since the integral ( f ( x ) = = f ( x ) dx is defined in ...
Page 43
... Pr ( a < X1 < b ) Pr ( c < X2 < d ) for every a < b and c < d , where a , b , c , and d are constants . PROOF . From the stochastic independence of X1 and X2 , the joint p.d.f. of X1 and X2 is ƒ1 ( x1 ) ƒ2 ( X2 ) . Accordingly , in the ...
... Pr ( a < X1 < b ) Pr ( c < X2 < d ) for every a < b and c < d , where a , b , c , and d are constants . PROOF . From the stochastic independence of X1 and X2 , the joint p.d.f. of X1 and X2 is ƒ1 ( x1 ) ƒ2 ( X2 ) . Accordingly , in the ...
Page 46
... Pr ( a1 < X1 < b1 ) Pr ( a2 < X2 < b2 ) · Pr ( an < Xn < bn ) [ [ Pr ( a ; < X ; < b ; ) n where the symbol ÏÏ¿ ( î ) is defined to be ... n ÏÏ ¢ ( i ) = 6 ( 1 ) 4 ( 2 ) ...... . p ( n ) . ... The theorem that E [ u ( X1 ) v ( X2 ) ] = E ...
... Pr ( a1 < X1 < b1 ) Pr ( a2 < X2 < b2 ) · Pr ( an < Xn < bn ) [ [ Pr ( a ; < X ; < b ; ) n where the symbol ÏÏ¿ ( î ) is defined to be ... n ÏÏ ¢ ( i ) = 6 ( 1 ) 4 ( 2 ) ...... . p ( n ) . ... The theorem that E [ u ( X1 ) v ( X2 ) ] = E ...
Contents
CHAPTER | 1 |
TRANSFORMATION OF VARIABLES cont | 17 |
CHAPTER | 27 |
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Common terms and phrases
A₁ A₂ best critical region binomial distribution c₁ cent confidence interval chi-square distribution composite hypothesis conditional p.d.f. confidence interval Consider continuous type critical region degrees of freedom denote a random discrete type distribution function distribution having p.d.f. distribution with mean EXAMPLE Exercises function of Y₁ hypothesis H₁ independent random variables integral Jacobian joint p.d.f. joint sufficient statistics Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent Mx(t My(t normal distribution n(x null hypothesis null simple hypothesis one-to-one transformation order statistics p.d.f. of Y₁ p₁ Poisson distribution positive integer Pr(a Pr(X Pr(Y probability density functions r₁ random experiment random sample random variables X1 Show significance level statistical hypothesis stochastically independent random theorem type of random unbiased statistic values variance o² X₁ X1 and X2 X₂ Xn denote Y₂ zero elsewhere μ₁ σ²