## Introduction to Mathematical StatisticsAn exceptionally clear and impeccably accurate presentation of statistical applications and more advanced theory. Included is a chapter on the distribution of functions of random variables as well as an excellent chapter on sufficient statistics. More modern technology is used in considering limiting distributions, making the presentations more clear and uniform. |

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Page 38

Robert V. Hogg. Clearly, if x < 0, then F(x) = 0; and if x > 1 , then F(x) = 1 . Thus

we can write F(x) = 0, = x\ = 1, x < 0, 0 < X < 1, 1 < JC. Recall, in the discrete case,

we had a function /that was associated with F through the

Robert V. Hogg. Clearly, if x < 0, then F(x) = 0; and if x > 1 , then F(x) = 1 . Thus

we can write F(x) = 0, = x\ = 1, x < 0, 0 < X < 1, 1 < JC. Recall, in the discrete case,

we had a function /that was associated with F through the

**equation**...Page 94

(1) If both members of

bE(X), or n2 = a + bnu (2) where /i, = E{X) and ^2 = E(Y). If both members of

aE(X) + ...

(1) If both members of

**Equation**(1) are integrated on x, it is seen that E(Y) = a +bE(X), or n2 = a + bnu (2) where /i, = E{X) and ^2 = E(Y). If both members of

**Equation**(1) are first multiplied by x and then integrated on x, we have E(XY) =aE(X) + ...

Page 381

But a heuristic argument can be made by solving for § — 9 to obtain d[ln L(9)) n

n_ 89 _ Z 62[\n L(9)] d2[\n L(9)] ' de2 ~ ee2 Let us rewrite this

80 I Since Z is the sum of the i.i.d. random variables d\nf(Xr, 9) . — , i= 1,2, ...,« ...

But a heuristic argument can be made by solving for § — 9 to obtain d[ln L(9)) n

n_ 89 _ Z 62[\n L(9)] d2[\n L(9)] ' de2 ~ ee2 Let us rewrite this

**equation**as \lnl(9) n80 I Since Z is the sum of the i.i.d. random variables d\nf(Xr, 9) . — , i= 1,2, ...,« ...

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Accordingly approximate best critical region chi-square distribution complete sufficient statistic conditional p.d.f. conditional probability confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random depend upon 9 discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters equation estimator of 9 Example Exercise F-distribution gamma distribution given H0 is true hypothesis H0 independent random variables integral joint p.d.f. Let the random Let Xu X2 limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Xu percent confidence interval Poisson distribution positive integer probability density functions probability set function quadratic form random experiment random sample random variables Xx reject H0 respectively sample space Section Show significance level simple hypothesis statistic for 9 sufficient statistic testing H0 theorem unbiased estimator variance a2 Xx and X2 Yu Y2 zero elsewhere