STAT7055 Semester 2 2022 Topic 3 Tutorial Questions
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STAT7055 Semester 2 2022
Topic 3 Tutorial Questions
1. Suppose the random variable X has the following probability distribution:
|
x |
0 |
1 |
2 |
|
p(x) |
0.6 |
0.3 |
0.1 |
Calculate the following (do it manually first, then try doing it in R using the sum func- tion):
(a) E(X)
(b) V (X)
(c) E !^X"
(d) V !^X"
2. The joint probability distribution of X and Y is shown in the following table:
|
1 |
x 2 |
3 |
||
|
y |
1 2 |
0.42 0.28 |
0.12 0.08 |
0.06 0.04 |
(a) Calculate P({X > 1}|{Y < 2}).
(b) Calculate the correlation coefficient between X and Y .
(c) Are X and Y independent?
(d) Determine the probability distribution of 
+ 
.
3. A new surgical procedure is said to be successful 80% of the time. Suppose that the operation is performed five times, with the results being independent of each other. We are interested in the successfulness of this operation.
(a) Define an appropriate random variable X for this problem.
(b) What particular type of distribution does the random variable X have? State the
values of the parameters.
Find the following probabilities:
(c) Exactly three operations are successful.
(d) Fewer than two operations are successful.
(e) More than three are successful.
4. A home security system is designed to have a 98% reliability rate. That is, in the event of a true burglary attempt, there is a 98% chance that the alarm will be triggered. Suppose that nine homes equipped with the system experience attempted burglaries. Find the probability that:
(a) At least one alarm is triggered.
(b) More than seven of the alarms are triggered.
5. It is recommended that women over 40 have a mammogram annually. A recent report indicated that if a woman has annual mammograms over a 10-year period, there is a 60% probability that there will be at least one false-positive result. If the annual test results are independent, what is the probability that in any one year a mammogram will produce a false-positive result?
6. A student takes an examination consisting of two multiple choice questions. Each mul- tiple choice question has four possible answers with only one correct. For the first question, suppose the probability that he knows the answer is 0.8 and the probability that he guesses is 0.2. If he guesses, he randomly chooses one of the four possible an- swers. For the second question, suppose the student gains confidence if he knew the answer to the first question. Specifically, if he knew the answer to the first question, then for the second question the probability that he knows the answer becomes 0.9 and the probability that he guesses becomes 0.1. If he does not know the answer to the first question, then for the second question the probability that he knows the answer and the probability that he guesses remain 0.8 and 0.2, respectively. Letting X denote the number of questions he gets correct, calculate the probability distribution of X and calculate the expected value of X .
7. Discussion Question
Find an example of two independent discrete random variables for which their sum and
difference have the same spread. That is, V (X + Y) = V (X − Y) = V (X) + V (Y).
8. swirl
Work through lessons 5 (note that we will cover the normal distribution in Topic 4) and 6 of the R Programming course.
2022-08-31