Introduction to Mathematical Statistics |
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Page 124
... determine the p.d.f. of Y = X2 . 4.19 . Let X1 , X2 be a random sample from the normal distribution n ( 0 , 1 ) . Show that the marginal p.d.f. of Y1 X1 / X2 is the Cauchy p.d.f. 1 = -∞ < Y1 < ∞ . g1 ( 31 ) = - π ( 1+ y2 ) ' Hint ...
... determine the p.d.f. of Y = X2 . 4.19 . Let X1 , X2 be a random sample from the normal distribution n ( 0 , 1 ) . Show that the marginal p.d.f. of Y1 X1 / X2 is the Cauchy p.d.f. 1 = -∞ < Y1 < ∞ . g1 ( 31 ) = - π ( 1+ y2 ) ' Hint ...
Page 158
... Determine a 90 per cent confidence interval for μ . = = 5.9 . Let two independent random samples , each of size 10 , from two independent normal distributions n ( μ1 , o2 ) and n ( μ2 , o2 ) yield ≈ = 4.8 , s1 = 8.64 , ÿ 5.6 , s2 ...
... Determine a 90 per cent confidence interval for μ . = = 5.9 . Let two independent random samples , each of size 10 , from two independent normal distributions n ( μ1 , o2 ) and n ( μ2 , o2 ) yield ≈ = 4.8 , s1 = 8.64 , ÿ 5.6 , s2 ...
Page 250
... determine max R ( 0 , w ) if it exists . 0 9.14 . Let X1 , X2 , ... , X , denote a random sample from a distribution n that is n ( μ , 0 ) , 0 < 0 < ∞ , where μ is unknown . Let Y = † ( X , – X ) 2 / n = = = S2 and let [ 0 , w ( y ) ...
... determine max R ( 0 , w ) if it exists . 0 9.14 . Let X1 , X2 , ... , X , denote a random sample from a distribution n that is n ( μ , 0 ) , 0 < 0 < ∞ , where μ is unknown . Let Y = † ( X , – X ) 2 / n = = = S2 and let [ 0 , w ( y ) ...
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A₁ A₂ Accordingly best critical region c₁ cent confidence interval chi-square distribution complete sufficient statistic compute conditional p.d.f. confidence interval Consider continuous type critical region decision function defined degrees of freedom denote a random discrete type distribution function distribution having p.d.f. Equation Example EXERCISES function F(x given H₁ hypothesis H independent random variables integral joint p.d.f. k₁ Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral normal distribution order statistics p.d.f. of Y₁ P(A₁ p₁ Poisson distribution positive integer probability density functions quadratic form random experiment random interval random sample random variables X1 respectively sample space Show significance level simple hypothesis statistic Y₁ stochastically independent random sufficient statistic theorem unbiased statistic variance o² w₁ X₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ