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Uppsala University
Department for Information Science
Statistics
B1
2012-10-02
Revised
WRITTEN EXAM FOR THE COURSE, PROBABILITY THEORY AND
STATISTICAL INFERENCE, B1 (7.5 ECTS)
Writing time: 1300-1700
Permitted aids:
Formulas for the course Probability Theory and Statistical Inference
Math-Handout, Lars Forsberg
Pocket calculator
Dictionary (or word-list)
Notations in the permitted aids are not allowed.
The written examination has 5 problems, for a total of 100 points.
If you desire clarification regarding the test, especially the wording of a problem, then
please alert an examination proctor. The examination proctors can contact the responsible
instructor.
After turning in your test, you may keep the test-pages with the question-statements.
INSTRUCTIONS:
A.
Carefully follow the instructions that are listed on the examination-directions page.
B.
State the assumptions that must be made for the method to be applicable.
C. Account for every essential step in your solution. If special concerns are raised in the
problem statement, then your solution must carefully address those concerns.
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Task 1. (10p)
The following is given: P(A) = 0.4, P(B) =0.4, P(C)=0.2, P(C∩A) = 0.2 and P(C∩B)= 0. Also
assume the events A and B to be independent.
Find the following probabilities:
a) P(AUBUC)
̅̅̅̅̅̅ )
b) P(A|𝐵𝑈𝐶
Task 2 (10)
Assume that 𝑌1 , 𝑌2 , 𝑌3 , 𝑌4 , 𝑌5 and 𝑌6 are independent and normally distributed with zero mean and
variances equal to one.
a) How is the following statistic distributed:
√2(𝑌1 + 𝑌2 )
2
2
√(𝑌1 − 𝑌1 + 𝑌2 ) + (𝑌2 − 𝑌1 + 𝑌2 )2
2
2
b) Find the probability
𝑌12 + 𝑌22 + 𝑌32
𝑃( 2
> 9.28)
𝑌4 + 𝑌52 + 𝑌62
Task 3. (30)
Given the following density for the random variable Y
𝑦
𝑓(𝑦) = { 8
0
𝑓𝑜𝑟 0 ≤ 𝑦 ≤ 4
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
a) Find the distribution function for Y.
b) Find P(0.2 < Y < 2.5).
c) Find the mean for Y.
d) Find the median for Y.
Task 4. (20)
Let Y1 ,Y2 ,Yn denote a random sample from the Poisson distribution with mean λ.
𝑝(𝑦) =
a)
b)
c)
d)
𝜆𝑦 𝑒 −𝜆
𝑦!
, y = 0, 1, 2, 3,….
Present the likelihood function.
Present the log likelihood function.
Find the ML estimator for λ.
The variance of the estimator can be obtained by
1
.
𝐸(−𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑔 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛)
Use this formula to find the variance of the estimator in task c) . Do you find what you
expect? Please justify your answer.
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Task 5. (30p)
Assume that you have a random sample of n shoppers and want to test the null hypothesis
that one-half of all shoppers prefer brand A to brand B , against the alternative that the
population proportion is less than one-half.
A. Let Y be equal to the number of shoppers that prefer brand A to brand B in a
sample of n = 10 shoppers. Define the rejection region for a test with a significance
level of approximately 5%.
B. For the test in task A calculate the probability for the Type two error if in fact only
40% of all shoppers prefer brand A to brand B.
C. Say that you in a sample of 390 shoppers find that Y = 175. Use these data to test
the hypothesis on the 5% significance level.
D. For the test in task C calculate the probability for the type two error if in fact only
40% of all shoppers prefer brand A to brand B.
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