Math 4600 Homework 8
1. Consider a process of molecules leaving a cell, independent of each other,
with a constant probabilitic rate of 2 per second.
a) Find probability that at least 2 molecules leave during a 4 second experiment?
b) How many molecules do you expect to leave, on average, during a 10 second
experiment?
c) What is the probability that the number of molecules that leave during the
10 second experiment is within 1 from the expected number?
d) What is the expected waiting time between second molecule that leaves and
the third one?
e) What is he probability that you have to wait longer than one second for the
first molecule to leave?
f) If you watch for a while and the molecules keep leaving and nothing enters
the cell, come up with a reason for why the process may stop being Poisson?
2. (computing) Generate times of 100 events of a Poisson process with λ
= 1. (To help: the command t=-(1/lambda)*log(1-rand(m,n)) will generate m
by n matrix of random numbers that are drawn from exponential distribution).
a) Plot the event times as dots in a line at the appropriate times (horizontal
axis should be time, dots, located on the same horizontal line, mark times of
events). Look at different parts of the array. Does it look random? (to check
that you did this correctly your last event should happen at t near 100.)
b) Count the number of events up to time 10. Is it close to the number we
expect from theory? Do the same thing up to time 50. Is it close this time?
c) Now look at your generated array and pretend that you have forgotten λ that
you used. Estimate it.
3. (computing) Conisder two populations of fruit flies. The first one has
mutation rate of 0.002 per generation and the second one has a mutation rate
of 0.0004 of generations. The results of experimentally measuring the number
of mutation differences in flies subpopulations are summarized in the following
diagram
0.09
0.08
0.07
Probability
0.06
0.05
0.04
0.03
0.02
0.01
0
−10
0
10
20
D
30
40
50
There is a debate in a newspaper in which one side claims that these flies
have been independently evolving for 1 million years and the other side claims
that have been evolving for 10 thousand years. Which do you think is more
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likely based on comparing the predicted distributions of mutation differences
with the data?
4. Let us say that the drug resistance in HIV results from a particular 3
base mutation. Find probability that the drug resistance develops in the HIV
population in one day. Do and explain all steps, even those that we did in class.
5. (From “Modeling the dynamics of Life”) Use the normal approximation
to binomial or poisson distribution (as appropriate) to find the follwing probabilities. Do not forget the continuity correction.
a) Thirty percent of cells in a small organism are not functioning. What is
the probability that an organ consisting of 250 cells is functioning if it requires
170 cells to work?
b) Ten percent of people are known to carry a certain gene In a sample of
200, what is the probability that between 5 and 15 carry the gene?
c) Molecule enters a cell at a rate of 0.045 per second. What is the probability
that between 150 and 200 (inclusive) enter during an hour?
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