## Introduction to Mathematical Statistics |

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Page 25

T of ro The ratio test may be used to

Thus there does not exist a positive number h such that Mx(t) exists for —h 3 t < h.

Accordingly the distribution having the p.d. f. f(z) of this example does not have ...

T of ro The ratio test may be used to

**show**that this series converges only if t < 0.Thus there does not exist a positive number h such that Mx(t) exists for —h 3 t < h.

Accordingly the distribution having the p.d. f. f(z) of this example does not have ...

Page 31

2.4. The moment-generating function of a random variable X is [% + (34)e']”.

-'No,.

2.4. The moment-generating function of a random variable X is [% + (34)e']”.

**Show**that 5 2. 9–2 ro-2'-x < *-āo;()() z-l (35. Let X have a binomial distribution.**Show**that *(·ft) = 0 *(o) = 1 vonp(1-p) y Vnp(1 − p) 2 _X-np_ —*— 1-2 o / M.(#)]-la-'No,.

Page 104

with mean 6, -oo 3 0 < co, and known variance a' is a sufficient statistic for 6. 5.14.

Let X1, X2, ..., Xn be a random sample from the normal distribution n(z; 0, 0), 0 <

0 ...

**Show**that the mean X of a random sample of size n from a normal distributionwith mean 6, -oo 3 0 < co, and known variance a' is a sufficient statistic for 6. 5.14.

Let X1, X2, ..., Xn be a random sample from the normal distribution n(z; 0, 0), 0 <

0 ...

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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