北大光华统计学考研资料-新祥旭光华辅导班

1、 北大光华统计学考研资料-新祥旭光华辅导班Exercises1. About MLE1-1. SAlso ,suppose that the value of is unknown (.a) Show that the MLE of does not exist.b)1-2.Also, suppose that the value of is unknown (. Find the MLE of 1-3. SAlso, suppose that the value of is unknown . Find the MLE of 1-4. Suppose that the lifetime of

2、 a certain type of lamp has an exponential distribution for which the value of the parameter is unknown. A random sample of n lamps of this type are tested for a period of T hours and the number X of lamps that fail during this period is observed; but the times at which the failures occurred are not

3、 noted. Determine the MLE of based on the observed value of X1-5. Suppose that form a random sample from a uniform distribution on the interval a,b, where both endpoints a and b are unknown. Find the MLE of the mean of the distribution.1-6. Suppose that form a random sample from a normal distributio

4、n for which both the mean and the variance are unknown. Find the MLE of the 0.95 quantile of the distribution. That is, of the point such that 1-7. Suppose that form a random sample of size n from a uniform distribution on the interval , where the value of is unknown. Show that the sequence of MLEs

5、of is a consistent sequence.2. About Estimation2-1. Suppose that a random sample is to be taken from a Bernoulli distribution for which the value of the parameter p is unknown. Suppose also that it is believed that the value of p is in the neighborhood of 0.2. How large a random sample must be taken

6、 in order that when p=0.2?For the conditions in the above exercise,use the central limit theorem to find approximately the size of a random sample that must be taken in order that when p=0.2?For the conditions in the above exercise, how large a random sample must be taken in order that when p=0.2?2-

7、2. Suppose that the random variables are independent, and each random variable has a continuous d.f. . Also, let the random variable Y be defined by the relation . Show that Y has a distribution with 2n degrees of freedom.2-3. Suppose that form a random sample from a normal distribution with mean an

8、d variance . Find the distribution of 2-4. Suppose that form a random sample from a normal distribution with mean and variance , and let denote the sample variance. Determine the smallest values of n for which the following relations are satisfied.a)b)2-5. Suppose that form a random sample from an e

9、xponential distribution with unknown mean . Describe a method for constructiong a confidence interval for with a specified confidence coefficient (0<.2-6. In the June 1986 issue of Consumer Reports, some data on the calorie content of beef hot dogs is given. Here are the numbers of calories in 20

10、 different hot dog brands:186,181,176,149,184,190,158,139,175,148,152,111,141,153,190,157,131,149,135,132Assume that these numbers are the observed values from a random sample of twenty independent normal random variables with mean and variance , both unknown. Find a 90 percent confidence interval for the mean number of calories.2-7. Suppose that a random variable X has a Poisson distribution for which the mean is unknown (. Find a statistic that will be an unbiased estimator of .2-8. Suppose that form a random