## Introduction to Mathematical Statistics |

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Page 46

Xn are said to be

) = f(z)f(x2) ... f.(c.). It follows immediately from this definition of the

Oln ...

Xn are said to be

**mutually stochastically independent**if, and only if, f(zi, a 2, ..., zn) = f(z)f(x2) ... f.(c.). It follows immediately from this definition of the

**mutual****stochastic independence**of X1, X2, ..., Xn that Pr(al < X1 < bi, a2 < X2 < b2, • '',Oln ...

Page 50

The reader should extend this proposition to the sum of n

z) ...

The reader should extend this proposition to the sum of n

**mutually stochastically****independent**chi-square variables. EXAMPLE 4. Let each of the**mutually****stochastically independent**random variables X1, X2,..., Xn have the same p.d.f. f(z) ...

Page 53

Hint: Write My(t) = E[e”)] and make use of the stochastic independence of X1 and

X2. 3.13. Let Y be the sum of n

variables X1, X2, ..., Xn with ri, r2, ..., rn degrees of freedom, respectively.

Hint: Write My(t) = E[e”)] and make use of the stochastic independence of X1 and

X2. 3.13. Let Y be the sum of n

**mutually stochastically independent**chi-squarevariables X1, X2, ..., Xn with ri, r2, ..., rn degrees of freedom, respectively.

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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