## Introduction to Mathematical Statistics |

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Page 41

For example, suppose that f(

of ...

For example, suppose that f(

**x1**,**x2**) is the p.d.f. of two random variables**X1 and****X2**. Then Y = X1 + X2 and Z =**X1**,**X2**are functions of the random variables**X1****and X2**, and we may be interested in finding the distribution of Y or Z. As a matterof ...

Page 46

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

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 94

These considerations actually motivate the definition of stochastic independence

of

conditional p.d.f. and a conditional expectation. Let f(

of ...

These considerations actually motivate the definition of stochastic independence

of

**X1 and X2**, page 42. We now proceed to generalize the definitions of aconditional p.d.f. and a conditional expectation. Let f(

**x1**,**x2**, ..., z) be the joint p.d.f.of ...

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### Common terms and phrases

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