cstrot Exercises 229 (a) A shipment of 1000 fuses contains 3 defective fuses. A sample of 25 fuses contained 0 defectives. (b) The speed of 100 vehicles was monitored. It was found that 63 vehicles exceeded the posted speed limit. (c) A telephone poll of registered voters one week before a statewide election showed that 48% would vote for the current governor, who was running for reelection. The final election returns showed that the incumbent won with 52% of the votes cast. Let X1,X2,X3, X4 be i.i.d. observations from a distribution with mean u and variance o2. 6.2 Consider the following four estimators of u: X2+ X3 30.1X10.2X2 +0.3X3 + 0.4X4, p4 = X. 2 (a) Show that all four estimators are unbiased. (b) Calculate the variance of each estimator. Which one has the smallest variance? (c) More generally, for a random sample of size n, show that if an estimator 01X1+a2X2 + +anXn, where a1, a2,... , an are constants, is unbiased, then its variance is minimum when a1 a2=.= to ai 1, af is minimized by choosing a1 a2=. = an = 1/n, i.e., f = X. (Hint: Subject = an 1/n.) Let X1,X2, . . .,Xn be a random sample from a U[0, e] distribution. Use the result of Exercise 5.44 to show the following. 0.3 (a) Xmax is a biased estimator of 0. What is its bias? (b) Xmin +Xmax is an unbiased estimator of 0. From this it follows that the midrange, defined as (Xmin + Xmax)/2, is an unbiased estimator of 0/2, which is the mean of the U[0, 0] distribution. 6.4 Let X1,X2, ...,Xn be a random sample from a distribution with mean u and variance a2, Show that X2 is a biased estimator of u2. What is its bias? (Hint: E(X2) - u2 Var(X) o2/n.) 6.5 Suppose we have n independent Bernoulli trials with true success probability p. Consider two estimators of p: p1 p where p is the sample proportion of successes and p2 1/2, a fixed constant.
cstrot Exercises 229 (a) A shipment of 1000 fuses contains 3 defective fuses. A sample of 25 fuses contained 0 defectives. (b) The speed of 100 vehicles was monitored. It was found that 63 vehicles exceeded the posted speed limit. (c) A telephone poll of registered voters one week before a statewide election showed that 48% would vote for the current governor, who was running for reelection. The final election returns showed that the incumbent won with 52% of the votes cast. Let X1,X2,X3, X4 be i.i.d. observations from a distribution with mean u and variance o2. 6.2 Consider the following four estimators of u: X2+ X3 30.1X10.2X2 +0.3X3 + 0.4X4, p4 = X. 2 (a) Show that all four estimators are unbiased. (b) Calculate the variance of each estimator. Which one has the smallest variance? (c) More generally, for a random sample of size n, show that if an estimator 01X1+a2X2 + +anXn, where a1, a2,... , an are constants, is unbiased, then its variance is minimum when a1 a2=.= to ai 1, af is minimized by choosing a1 a2=. = an = 1/n, i.e., f = X. (Hint: Subject = an 1/n.) Let X1,X2, . . .,Xn be a random sample from a U[0, e] distribution. Use the result of Exercise 5.44 to show the following. 0.3 (a) Xmax is a biased estimator of 0. What is its bias? (b) Xmin +Xmax is an unbiased estimator of 0. From this it follows that the midrange, defined as (Xmin + Xmax)/2, is an unbiased estimator of 0/2, which is the mean of the U[0, 0] distribution. 6.4 Let X1,X2, ...,Xn be a random sample from a distribution with mean u and variance a2, Show that X2 is a biased estimator of u2. What is its bias? (Hint: E(X2) - u2 Var(X) o2/n.) 6.5 Suppose we have n independent Bernoulli trials with true success probability p. Consider two estimators of p: p1 p where p is the sample proportion of successes and p2 1/2, a fixed constant.
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Part c of 6.2 only.
![cstrot Exercises
229
(a) A shipment of 1000 fuses contains 3 defective fuses. A sample of 25 fuses contained
0 defectives.
(b) The speed of 100 vehicles was monitored. It was found that 63 vehicles exceeded
the posted speed limit.
(c) A telephone poll of registered voters one week before a statewide election showed
that 48% would vote for the current governor, who was running for reelection. The
final election returns showed that the incumbent won with 52% of the votes cast.
Let X1,X2,X3, X4 be i.i.d. observations from a distribution with mean u and variance o2.
6.2
Consider the following four estimators of u:
X2+ X3
30.1X10.2X2 +0.3X3 + 0.4X4, p4 = X.
2
(a) Show that all four estimators are unbiased.
(b) Calculate the variance of each estimator. Which one has the smallest variance?
(c) More generally, for a random sample of size n, show that if an estimator
01X1+a2X2 + +anXn, where a1, a2,... , an are constants, is unbiased, then its
variance is minimum when a1 a2=.=
to ai 1, af is minimized by choosing a1 a2=.
= an = 1/n, i.e., f = X. (Hint: Subject
= an 1/n.)
Let X1,X2, . . .,Xn be a random sample from a U[0, e] distribution. Use the result of
Exercise 5.44 to show the following.
0.3
(a) Xmax is a biased estimator of 0. What is its bias?
(b) Xmin +Xmax is an unbiased estimator of 0. From this it follows that the midrange,
defined as (Xmin + Xmax)/2, is an unbiased estimator of 0/2, which is the mean of
the U[0, 0] distribution.
6.4 Let X1,X2, ...,Xn be a random sample from a distribution with mean u and variance
a2, Show that X2 is a biased estimator of u2. What is its bias? (Hint: E(X2) - u2
Var(X) o2/n.)
6.5 Suppose we have n independent Bernoulli trials with true success probability p. Consider
two estimators of p: p1 p where p is the sample proportion of successes and p2 1/2,
a fixed constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f1ebd08-7e04-46cf-88d9-0b90774940c7%2F52368da6-2f5e-4f6e-b6ff-983d2f30648a%2F45qaus7.jpeg&w=3840&q=75)
Transcribed Image Text:cstrot Exercises
229
(a) A shipment of 1000 fuses contains 3 defective fuses. A sample of 25 fuses contained
0 defectives.
(b) The speed of 100 vehicles was monitored. It was found that 63 vehicles exceeded
the posted speed limit.
(c) A telephone poll of registered voters one week before a statewide election showed
that 48% would vote for the current governor, who was running for reelection. The
final election returns showed that the incumbent won with 52% of the votes cast.
Let X1,X2,X3, X4 be i.i.d. observations from a distribution with mean u and variance o2.
6.2
Consider the following four estimators of u:
X2+ X3
30.1X10.2X2 +0.3X3 + 0.4X4, p4 = X.
2
(a) Show that all four estimators are unbiased.
(b) Calculate the variance of each estimator. Which one has the smallest variance?
(c) More generally, for a random sample of size n, show that if an estimator
01X1+a2X2 + +anXn, where a1, a2,... , an are constants, is unbiased, then its
variance is minimum when a1 a2=.=
to ai 1, af is minimized by choosing a1 a2=.
= an = 1/n, i.e., f = X. (Hint: Subject
= an 1/n.)
Let X1,X2, . . .,Xn be a random sample from a U[0, e] distribution. Use the result of
Exercise 5.44 to show the following.
0.3
(a) Xmax is a biased estimator of 0. What is its bias?
(b) Xmin +Xmax is an unbiased estimator of 0. From this it follows that the midrange,
defined as (Xmin + Xmax)/2, is an unbiased estimator of 0/2, which is the mean of
the U[0, 0] distribution.
6.4 Let X1,X2, ...,Xn be a random sample from a distribution with mean u and variance
a2, Show that X2 is a biased estimator of u2. What is its bias? (Hint: E(X2) - u2
Var(X) o2/n.)
6.5 Suppose we have n independent Bernoulli trials with true success probability p. Consider
two estimators of p: p1 p where p is the sample proportion of successes and p2 1/2,
a fixed constant.
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