Introduction to Mathematical Statistics |
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Page 53
... Let X1 and Y = X1 X2 have chi - square distributions with r1 and r degrees of freedom , respectively . Here r1 < r . Show that X2 has a chi - square distribution with r1 degrees of freedom . Hint : Write My ( t ) Elet ( X + X ) ] and ...
... Let X1 and Y = X1 X2 have chi - square distributions with r1 and r degrees of freedom , respectively . Here r1 < r . Show that X2 has a chi - square distribution with r1 degrees of freedom . Hint : Write My ( t ) Elet ( X + X ) ] and ...
Page 73
... Let X1 and X2 be stochastically independent random variables with joint p.d.f. ƒ1 ( x1 ) ƒ2 ( x2 ) that is positive on the two - dimensional space A. Let Y1 = u1 ( X1 ) , a func- tion of X1 alone , and Y2 u2 ( X2 ) , a function of X2 ...
... Let X1 and X2 be stochastically independent random variables with joint p.d.f. ƒ1 ( x1 ) ƒ2 ( x2 ) that is positive on the two - dimensional space A. Let Y1 = u1 ( X1 ) , a func- tion of X1 alone , and Y2 u2 ( X2 ) , a function of X2 ...
Page 101
... X1 + X2 + X3 is a sufficient statistic for 0. For convenience , let X3 and Y3 X3 , so by the corresponding transformation the space = = that Yı Y2 Y2 = X2 { ( X1 , X2 , X3 ) , 0 < x < ∞ is mapped into the space ( y1 , 32 , Ys ) ; 0 ...
... X1 + X2 + X3 is a sufficient statistic for 0. For convenience , let X3 and Y3 X3 , so by the corresponding transformation the space = = that Yı Y2 Y2 = X2 { ( X1 , X2 , X3 ) , 0 < x < ∞ is mapped into the space ( y1 , 32 , Ys ) ; 0 ...
Contents
CHAPTER | 1 |
TRANSFORMATION OF VARIABLES cont | 17 |
CHAPTER | 27 |
Copyright | |
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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