Introduction to Mathematical Statistics |
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Page 46
... ( x1 , x2 , ... , Xn ) = ƒ1 ( X1 ) ƒ2 ( X2 ) ··· fn ( n ) . It follows immediately from this definition of the mutual ... random variables X1 and X2 becomes , for mutually sto- chastically independent random variables X1 , X2 , ··· , Xn ...
... ( x1 , x2 , ... , Xn ) = ƒ1 ( X1 ) ƒ2 ( X2 ) ··· fn ( n ) . It follows immediately from this definition of the mutual ... random variables X1 and X2 becomes , for mutually sto- chastically independent random variables X1 , X2 , ··· , Xn ...
Page 47
... variables X1 and X2 , show that X1 and X2 are stochasti- cally independent and that E [ e “ ( X1 + X , ) ] = ( 1 ... random variables X1 and X2 , provided −1 < p < 1. Find the marginal probability density functions fi ( x1 ) and ƒ2 ( x2 ) ...
... variables X1 and X2 , show that X1 and X2 are stochasti- cally independent and that E [ e “ ( X1 + X , ) ] = ( 1 ... random variables X1 and X2 , provided −1 < p < 1. Find the marginal probability density functions fi ( x1 ) and ƒ2 ( x2 ) ...
Page 53
Robert V. Hogg, Allen Thornton Craig. 8.12 . Let X1 and X , be stochastically independent random variables . Let X1 and Y = X1 X2 have chi - square distributions with r1 and r degrees of freedom , respectively . Here r1 < r . Show that ...
Robert V. Hogg, Allen Thornton Craig. 8.12 . Let X1 and X , be stochastically independent random variables . Let X1 and Y = X1 X2 have chi - square distributions with r1 and r degrees of freedom , respectively . Here r1 < r . Show that ...
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