## Introduction to mathematical statistics |

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Page 142

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 r —

degrees of ...

**Let X1**and X2 be stochastically independent random variables.**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 r —

degrees of ...

Page 143

X2 and Z = X\ + X\. Show that the moment- generating function of the joint

distribution of Y and Z is Elexpit^X, + X2) + t2(X\ + XI)]} = exp[*f/(1 - 2*,)] for — oo

< t1 ...

**Let X1**, X2 be a random sample from the normal distribution «(0, 1). Let Y = X1 +X2 and Z = X\ + X\. Show that the moment- generating function of the joint

distribution of Y and Z is Elexpit^X, + X2) + t2(X\ + XI)]} = exp[*f/(1 - 2*,)] for — oo

< t1 ...

Page 149

Let X and Y be stochastically independent random variables with means fi1, p2

and variances a2, a$. ...

distribution with mean p and variance a2. Show that E(S2) = (n — l)a2/n, where

S2 ...

Let X and Y be stochastically independent random variables with means fi1, p2

and variances a2, a$. ...

**Let X1**, X2, .... Xn be a random sample of size n from adistribution with mean p and variance a2. Show that E(S2) = (n — l)a2/n, where

S2 ...

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Accordingly best critical region binomial distribution cent confidence interval Chapter chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges stochastically critical region decision function defined degrees of freedom denote a random discrete type distribution having p.d.f. Equation Example EXERCISES F distribution function of Y1 given H0 is true independent random variables inequality integral joint p.d.f. Let the random Let X1 limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral order statistics Poisson distribution positive integer power function Pr X1 probability density functions probability set function quadratic form random experiment random interval random sample random variables X1 reject H0 respectively sample space Show significance level simple hypothesis H0 statistic for 9 statistic Y1 stochastically independent random subset testing H0 theorem type of random unbiased statistic variance a2 X1 and X2 Xn denote zero elsewhere