Introduction to Mathematical Statistics |
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Page 86
... ( t ) = p ′ [ 1 − ( 1 − p ) et ] - ' . Find the mean and the variance of this distribution . 3.13 . Let X1 and X2 ... Mx + 1 - αμκ − p ) dp αμκ dp p ) [ nkμ - 1 + dx ] Pk + 1 = p ( 1 − p ) [ nkμx Hint . Compute dμk / dp and dμx / dp . 3.2 ...
... ( t ) = p ′ [ 1 − ( 1 − p ) et ] - ' . Find the mean and the variance of this distribution . 3.13 . Let X1 and X2 ... Mx + 1 - αμκ − p ) dp αμκ dp p ) [ nkμ - 1 + dx ] Pk + 1 = p ( 1 − p ) [ nkμx Hint . Compute dμk / dp and dμx / dp . 3.2 ...
Page 88
... ( t ) = Σ etif ( x ) = Σ = e I -m Ž x = 0 e - memet = e for all real values of t . Since M ' ( t ) = and M " ( t ) = em ( et - 1 ) ( met ) , x = 0 ( met ) x x ! etx = em ( et − 1 ) mte - m x ! em ( et − 1 ) ( met ) + em ( et −1 ) ( met ) ...
... ( t ) = Σ etif ( x ) = Σ = e I -m Ž x = 0 e - memet = e for all real values of t . Since M ' ( t ) = and M " ( t ) = em ( et - 1 ) ( met ) , x = 0 ( met ) x x ! etx = em ( et − 1 ) mte - m x ! em ( et − 1 ) ( met ) + em ( et −1 ) ( met ) ...
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A₁ A₂ Accordingly c₁ chi-square distribution complete sufficient statistic compute conditional p.d.f. confidence interval 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 hypothesis H₁ independent random variables integral joint p.d.f. k₁ Let the random Let X1 Let Y₁ likelihood ratio limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent noncentral normal distribution order statistics p.d.f. of Y₁ P(A₁ Poisson distribution positive integer probability density functions probability set function 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 t₂ theorem unbiased statistic variance o² W₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ σ²