## Introduction to Mathematical Statistics |

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Page 53

Let X, and Y = XI + X2 have chi-square distributions with ri and r

r — ri

stochastic ...

Let X, and Y = XI + X2 have chi-square distributions with ri and r

**degrees of****freedom**, respectively. Here ri < r. Show that X2 has a chi-square distribution withr — ri

**degrees of freedom**. Hint: Write My(t) = E[e”)] and make use of thestochastic ...

Page 193

"Because the X; are mutually stochastically independent, Qisa” is the sum of a

mutually stochastically independent random variables, each having a chi-square

distribution with b – 1

"Because the X; are mutually stochastically independent, Qisa” is the sum of a

mutually stochastically independent random variables, each having a chi-square

distribution with b – 1

**degrees of freedom**. Hence Q/a” has a chi-square ...Page 194

It is easy to show (Exercise 10.1) that Q:/g” has a chi-square distribution with b(a

– 1)

that ically nt and that Q4/o” has a chisquare distribution with ab – 1 – boa - 1) = b

- 1 ...

It is easy to show (Exercise 10.1) that Q:/g” has a chi-square distribution with b(a

– 1)

**degrees of freedom**. Since Q. = ax (X. — X)' > 0, the theorem enables i-l sthat ically nt and that Q4/o” has a chisquare distribution with ab – 1 – boa - 1) = b

- 1 ...

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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