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The above results hold if sampling is from an infinite population or with replacement from a finite population. If sampling is without replacement from a finite population of size N, then the mean of the sampling distribution of variances is given by E(S2) As N S `, this reduces to (16). mS2 a N N 1 ba n n 1 bs2 (19)

By taking all possible random samples of size n drawn from a population and then computing the variance for each sample, we can obtain the sampling distribution of variances. Instead of finding the sampling distribution ^ of S2 or S2 itself, it is convenient to find the sampling distribution of the related random variable nS2 s2 (n 1)S2 s2

p p (b) From LC 2=!2 and L=C R2 we get L 2R=!0 and C 2=R!0 . Substituting L and C into (4a) gives 0 1 1 1 1 p p H s or H j! 4b 2 1 2 s=!0 s=!0 2 2 1 j 2 !=!0 !=!0 2 jHj2 1 1 4 1 !=!0 4 and tan 1 ! p 2!0 ! !2 !2 0

6.2 6.3 6.4 6.5 6.6 6.7 7 7.1 7.2 7.3

# X)2

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Note that H j! is independent of R. The network passes low-frequency sinusoids and rejects, or attenuates, the high-frequency sinusoids. It is a low-pass lter with a half-power frequency of ! !0 and, in this case, the p p magnitude of the frequency response is jH j!0 j jH 0 j= 2 2=4 and its phase angle is H j!0 =2. (c) For !0 1, H s 1 1 p 2 1 2s s2 1 1 jHj 4 1 !4

# X)2

(20)

H j!

The distribution of this random variable is described in the following theorem. Theorem 5-6 If random samples of size n are taken from a population having a normal distribution, then the sampling variable (20) has a chi-square distribution with n 1 degrees of freedom.

1 1 p 2 1 j 2! !2

Because of Theorem 5-6, the variable in (20) is often denoted by x2. For a proof of this theorem see Problem 5.22.

C.10

4c

In Theorems 5-4 and 5-5 we found that the standardized variable Z # X m s> !n (21)

is normally distributed if the population from which samples of size n are taken is normally distributed, while it is asymptotically normal if the population is not normal provided that n 30. In (21) we have, of course, assumed that the population variance s2 is known. It is natural to ask what would happen if we do not know the population variance. One possibility is to estimate the population variance by using the sample variance and then put the corresponding standard deviation in ^ (21). A better idea is to replace the s in (21) by the random variable S giving the sample standard deviation and to seek the distribution of the corresponding statistic, which we designate by T # X

p ! 2! 1 !2

(22)

The RC network of Fig 12-4 b was de ned as a rst-order low-pass lter with half-power frequency at !0 1=R1 C2 . The circuit of Fig. 12-13 is called a second-order Butterworth lter. It has a sharper cuto .

We can then show by using Theorem 4-6, page 116, that T has Student s t distribution with n 1 degrees of freedom whenever the population random variable is normally distributed. We state this in the following theorem, which is proved in Problem 5.24. Theorem 5-7 If random samples of size n are taken from a normally distributed population, then the statistic (22) has Student s distribution with n 1 degrees of freedom.

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