STA 130B Poisson Distribution for Statistics Hi, I don’t know how to solve this Question 6 , could someone help me to answer it ? Thx Homework 4 STA 130B:

STA 130B Poisson Distribution for Statistics Hi, I don’t know how to solve this Question 6 , could someone help me to answer it ? Thx Homework 4
STA 130B: Mathematical Statistics Brief Course
Due: Weds 3/6, in class
Problem 1. [Rice, Problem 6, pg. 363, revised]
Consider the coin-tossing experiment mentioned in class where there are two coins, Coin 0 and
Coin 1, with probability p of heads equal to 0.5 and 0.7, respectively. Professor Rajaratnam is
handed one of the coins (he does not know which one), which he tosses until a heads comes up.
The total number of tosses X is recorded, and based on this, we form a hypothesis test of the
following hypotheses:
H0 : p = 0.5
H1 : p = 0.7.
(a) What is the significance level of a test that rejects H0 if X ? 8? Hint: If X has a geometric
distribution, then P (X ? x) = 1 ? (1 ? p)x .
(b) What is the power of the test described in part (a)?
Problem 2. [Rice, Problem 19, pg. 364]
Under H0 , a random variable X has the cdf F0 = x2 , for 0 ? x ? 1. Under H1 , it has the cdf
F1 = x3 , for 0 ? x ? 1.
(a) Derive the likelihood ratio test of H0 vs. H1 .
(b) What is the rejection region of a level 0.05 test?
(c) What is the power of the test?
Problem 3. [Rice, Problem 17, pg. 364, revised]
Suppose that X1 , …, Xn ?iid N (0, ? 2 ), and consider testing H0 : ? = ?0 vs. H1 : ? = ?1 , where
?1 > ?0 .
(a) Derive the likelihood ratio test based on the data.
(b) Determine the value of c for the test you derived in part (a) for a level 0.01 test.
(c) Show that the test in part (a) is uniformly most powerful for testing H0 : ? = ?0 vs.
H1 : ? > ?0 .
Problem 4. [Monotone Likelihood Ratio]
(a) Suppose that X1 , …, XP
n ?iid Bernoulli(p). Show that the joint pmf has monotone likelihood
ratio in the statistic Y = ni=1 Xi .
(b) Suppose that X1 , …, Xn ?iid N (µ, ? 2 ), where ? 2 is known. Show that the joint pdf has
monotone likelihood ratio in the statistic X?.
(c) Suppose that X1 , …, Xn ?iid Gamma(?, ?), where ? is known. Show that the joint pdf has
monotone likelihood ratio in the statistic ?ni=1 Xi .
Problem 5. [Rice, Problem 12, pg. 363]
Let X1 , …, Xn be a sample from the exponential distribution. Derive a generalized likelihood
ratio test of H0 : ? = ?0 vs. H1 : ? 6= ?0 and show that the rejection region is of the form
{X? exp[??0 X?] ? c}.
Problem 6. [Buses make good Poisson examples]
The number of Unitrans buses passing through the Silo was recorded for 100 three-minute time
periods:
Number of Buses Count
0
10
1
35
2
35
3
20
(a) Use a goodness of fit test with ? = 0.05 to assess whether a Poisson distribution is a good
fit for the data.
(b) Suppose that we expect the ratios of the above cells to be 1:2:2:1, respectively. Use a goodness of fit test with ? = 0.05 to assess whether these expected ratios are consistent with the data.

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