## Introduction to Mathematical Statistics |

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Page 22

—oo if X is a continuous type of random variable, or E(ex) = XCet f(z), if X is a

discrete type of random variable. This expectation is called the

Mx(t) = E(ex).

—oo if X is a continuous type of random variable, or E(ex) = XCet f(z), if X is a

discrete type of random variable. This expectation is called the

**moment**-**generating function**of X (or of the distribution) and is denoted by Mx(t). That is,Mx(t) = E(ex).

Page 25

Then f(z) - r2:3' 2 = 1, 2, 3, e e - = 0 elsewhere, is the p.d.f. of a discrete type of

random variable X. The

is given by Mx(t) = E(e”) = XCeof (x) - co 6e. T of ro The ratio test may be used to ...

Then f(z) - r2:3' 2 = 1, 2, 3, e e - = 0 elsewhere, is the p.d.f. of a discrete type of

random variable X. The

**moment**-**generating function**of this distribution, if it exists,is given by Mx(t) = E(e”) = XCeof (x) - co 6e. T of ro The ratio test may be used to ...

Page 143

variable Y depends upon the positive integer n, in this chapter we shall write the

which is essentially Curtiss' modification of a theorem of Lévy and Cramér,

explains ...

variable Y depends upon the positive integer n, in this chapter we shall write the

**moment**-**generating function**of Y in the form My(t; n). The following theorem,which is essentially Curtiss' modification of a theorem of Lévy and Cramér,

explains ...

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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