## Introduction to Mathematical Statistics |

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Page 95

If X1 and X2 are random variables of the discrete type having p.d.f. f(zi, 22) - (zi +

2a-2)/18, (r1, 22) - (1, 1), (1, 2), (2, 1), (2, 2), Zero elsewhere, determine the

conditional mean and

Y, ...

If X1 and X2 are random variables of the discrete type having p.d.f. f(zi, 22) - (zi +

2a-2)/18, (r1, 22) - (1, 1), (1, 2), (2, 1), (2, 2), Zero elsewhere, determine the

conditional mean and

**variance**of X2, given X1 = 21, 21 = 1 or 2. 5.5. Let Y, & Y, &Y, ...

Page 96

+ X)/9 is normal with mean 6 and

statistic for 6. The statistic X1 is also normal with mean 6 and

X1 is also an unbiased statistic for 0. Although ox” < ox", we cannot say, with n =

9, ...

+ X)/9 is normal with mean 6 and

**variance**or” = 1/9. Thus X is an unbiasedstatistic for 6. The statistic X1 is also normal with mean 6 and

**variance**ax,” = 1; soX1 is also an unbiased statistic for 0. Although ox” < ox", we cannot say, with n =

9, ...

Page 214

This

multiplied by f(z) and integrated on 2, the result obtained will be non-negative.

This result is oo co 2 s s so – u2) — o - ofte y) dy da: -so so so-o-o-o-o-o-o: y) dy

da.

This

**variance**is non-negative and is at most a function of z alone. If then, it ismultiplied by f(z) and integrated on 2, the result obtained will be non-negative.

This result is oo co 2 s s so – u2) — o - ofte y) dy da: -so so so-o-o-o-o-o-o: y) dy

da.

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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