## Introduction to Mathematical Statistics |

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Page 17

Let f(z) = c/15, a = 1, 2, 3, 4, 5, zero elsewhere, be the p.a.f. of X. Find

2],

elsewhere, be the p.a.f. of X. If A1 = {z; 1 < z < 2} and A2 = {z; 4 × 2 × 5}, find P(A1\

J A2) ...

Let f(z) = c/15, a = 1, 2, 3, 4, 5, zero elsewhere, be the p.a.f. of X. Find

**Pr**[X = 1 or2],

**Pr**[J/3 < X → 5/2], and**Pr**[1 < X • 2]. 1.23. Let f(z) = 1/2”, 1 < x < co, zeroelsewhere, be the p.a.f. of X. If A1 = {z; 1 < z < 2} and A2 = {z; 4 × 2 × 5}, find P(A1\

J A2) ...

Page 38

If the random variable X has a chi-square distribution with r degrees of freedom,

then, with c1 < ca, we have, since

X 3 ci). To compute such a probability, we need the value of an integral like

If the random variable X has a chi-square distribution with r degrees of freedom,

then, with c1 < ca, we have, since

**Pr**(X = Cl) = 0,**Pr**(c. 3 X 3 c) =**Pr**(X 3 ca) —**Pr**(X 3 ci). To compute such a probability, we need the value of an integral like

**Pr**(x ...Page 43

If X1 and X2 are stochastically independent random variables with marginal

probability density functions f(x1) and f(z2), respectively, then

< d) =

are ...

If X1 and X2 are stochastically independent random variables with marginal

probability density functions f(x1) and f(z2), respectively, then

**Pr(a**3 X, < b, c < X2< d) =

**Pr(a**3 X, < b)**Pr**(c < X, < d) for every a < b and c < d, where a, b, c, and dare ...

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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