## Introduction to Mathematical Statistics |

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Page 29

(b) The

2=0 a ! converges, for all values of m, to e”. Consider the function f(z) defined by f(

x) = ** = - 0,1,2,... * = 0 elsewhere, where m > 0. Since m > 0, then f(z) > 0 and ...

(b) The

**Poisson distribution**. Recall that the series co 2 3 2: 1 + m + #4 #4 ... = 2* *2=0 a ! converges, for all values of m, to e”. Consider the function f(z) defined by f(

x) = ** = - 0,1,2,... * = 0 elsewhere, where m > 0. Since m > 0, then f(z) > 0 and ...

Page 30

That is, a

is frequently written 22-g f(x) = **, * = 0, 1,2,... = 0 elsewhere. ExAMPLE 3.

Suppose X has a

= *† ...

That is, a

**Poisson distribution**has u = 0° = m > 0. On this account, a Poisson p.d.f.is frequently written 22-g f(x) = **, * = 0, 1,2,... = 0 elsewhere. ExAMPLE 3.

Suppose X has a

**Poisson distribution**with u = 2. Then the p.d.f. of X is 22 –2 f(z)= *† ...

Page 144

Since there exists a distribution, namely, the

that has this moment-generating function e", then in accordance with the theorem

and under the conditions stated, it is seen that Y has a limiting

Since there exists a distribution, namely, the

**Poisson distribution**with mean u,that has this moment-generating function e", then in accordance with the theorem

and under the conditions stated, it is seen that Y has a limiting

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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