Download EECE 619: Introduction to Random Processes Homework 1: Given

EECE 619: Introduction to Random Processes
Homework 1: Given 10/07/10; Due 10/14/10
Do the following problems:
1. (10) A random variable Y is defined by Y = aX + b, where X is a standard Gaussian random
variable. Derive the distribution of Y using the characteristic function.
2. (10) Suppose X is a random variable with cdf FX (x). Let Y = g(X), where g(x) = x + 2 if
x < −2, g(x) = 0 if −2 ≤ x ≤ 2, and g(x) = x − 2 if x > 2. Find the cdf of Y .
3. (20) As scientists decode the human genome and study the significance of specific genes, it is
becoming clear that some genes are strongly linked to specific diseases. Thus, genetic tests
represent a potentially important way to test for these diseases. However, no gene is a perfect
predictor of a disease. In this problem, you are asked to assess the quality of a genetic test
for a disease, and decide whether the test is useful.
It has been determined that a gene M is found in 95% of people who have (or will get) disease
X, that is:
P (gene M | person has X) = 0.95
Also, the gene is not found in 95% of the people who do not have disease X, that is:
P (no gene M | person does not have X) = 0.95
It is also known that, in the population that the patients come from, the disease occurs in 10
of every 100,000 people. Suppose patient A is found to have the gene. What is the probability
that A has disease X?
Based on your analysis, would you consider this a good test? Justify your answer, and briefly
discuss why you see the results that you do and what they mean.
4. (10) A researcher wishes to study a 1 km × 1 km region in a remote area, and decides that the
best way to monitor it would be to place sensors all over the region of interest. She acquires
1,000,000 sensor devices and drops them randomly over the region from a helicopter, trying
to maintain a uniform density. To check the results, she marks out a 10 m × 10 m area in
the region and counts the actual number of sensors, M , in that area. What is the expected
value of M ? Suggest a reasonable distribution for M , and justify your suggestion. Based on
this, write an expression for the pmf of M .
(Note: This problem requires you to think a bit beyond what we have covered in class, but
it should not be too difficult. The distribution you suggest may only be an approximation.).
5. (10) We saw that, under certain circumstances, the binomial distribution can be approximated
by two other distributions that are simpler to work with. In each of the following cases, apply
both approximations, if possible, and compare the answers. Indicate which one you think is
closer to the correct answer. If an approximation is not possible, indicate why you think so.
You may use calculators or MATLAB programs for calculation.:
a) A measuring device makes errors with a probability of 0.001 on each measurement. What
is the probability of more than 20 errors in 10,000 measurements?
b) A difficult experiment has a probability 0.8 of succeeding each time it is done. What is
the probability of 98 successful experiments in 100 attempts?
6. (10) A team of scientists is planning to look for samples of a rare micro-organism poissonia
stochastica (p. stoch.) thought to have great potential in the treatment of certain diseases.
They have been offered $100,000 for their search by the Institute of Hopeless Causes. The
organism is found only near deep ocean hydrothermal vents where water temperatures can
reach as high as 450 degrees Celsius. The scientists have built a robot scoop that collects a
sample near the vents and brings it to the surface for analysis. However, collecting samples
near the vents is extremely dangerous, and the team wants to minimize the exposure of their
expensive robot. They have estimated that the probability of finding a p. stoch. in each
sample scoop is 0.002, and at least two live p. stoch. are required for their research.
a) Estimate how many samples the robot scoop must collect in order to have a 0.95 probability
of getting the two organism samples required.
b) Suppose that each collection by the robot costs $500. Given their funds, what is the
probability that the scientists will be able to achieve their goal of two p. stoch. samples.
Remember that these are dedicated scientists who do not spend any money on food or drink,
so all the funds go to the scientific mission.
Try to find answers that are as accurate as possible, but even an approximate answer is better
than none.