## Introduction to Mathematical Statistics |

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Page 86

Thus the joint p.d.f. of the three

middle in magnitude of X1, X2, X3; Ya = maximum of X1, X2, X3 is g(y1, y2, ya) =

|Jilf(yi)f(y)f(ys) + |J2|f(y)f(yi)f(ys) + ... + |Js|f(ys)f(y2)f(yi), a < y < y2 < ya < b, ...

Thus the joint p.d.f. of the three

**order statistics**Yi = minimum of X1, X2, X3; Y2 =middle in magnitude of X1, X2, X3; Ya = maximum of X1, X2, X3 is g(y1, y2, ya) =

|Jilf(yi)f(y)f(ys) + |J2|f(y)f(yi)f(ys) + ... + |Js|f(ys)f(y2)f(yi), a < y < y2 < ya < b, ...

Page 155

Finally the joint p.d.f. of any two

expressed in terms of F(z) and f(z). We have wo, m-soo's sos.sos now." j(yn) dyn “

” dy;41 dy;-1 ... dyiri dy, “' dyi–1. Since, for y > 0, s TFG) – Folosa) do -- roofer w _ [

F(u) ...

Finally the joint p.d.f. of any two

**order statistics**, say, Y, & Yi, is as easilyexpressed in terms of F(z) and f(z). We have wo, m-soo's sos.sos now." j(yn) dyn “

” dy;41 dy;-1 ... dyiri dy, “' dyi–1. Since, for y > 0, s TFG) – Folosa) do -- roofer w _ [

F(u) ...

Page 162

Let Y, 3 Y. 3 Ya K. Y., be the

distribution of the continuous type. The probability that the random interval (Yi, Y.)

includes the median £0.5 of the distribution will be computed. We have Pr(Y| ...

Let Y, 3 Y. 3 Ya K. Y., be the

**order statistics**of a random sample of size 4 from adistribution of the continuous type. The probability that the random interval (Yi, Y.)

includes the median £0.5 of the distribution will be computed. We have Pr(Y| ...

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Accordingly best critical region binomial distribution cent confidence interval chi-square distribution composite hypothesis conditional p.d.f. confidence interval Consider continuous type degrees of freedom denote a random discrete type distribution function distribution having p.d.f. distribution with mean ExAMPLE Exercises f(zi function F(z hypothesis H1 independent random variables integral Jacobian joint p.d.f. joint sufficient statistics Let the random Let X1 limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent Mx(t My(t normal distribution n(z null hypothesis null simple hypothesis one-to-one transformation order statistics p.d.f. of Yi Poisson distribution positive integer Pr(a Pr(X Pr(Y probability density functions quadratic form random experiment random interval random sample random variables X1 respectively ſ ſ sample space Show significance level ſº stochastically independent random subset sufficient statistic Yi theorem tion type of random unbiased statistic values X1 and X2 Xn denote zero elsewhere