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 the two populations of fruit flies that we talked
about in class. 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. HIV RNA consists of 104 bases. Suppose that new data give an estimate
of about 106 of new fully functional viral particles produced per day, and that
probability of an error during replication is 2 ∗ 10−5 per base per replication
cycle.
a) What is the total expected number of errors per replication cycle?
b) Find probability to get a mutant with mutation in exactly 1 base (use appropriate approximation as needed). What number of these mutants do you expect
to find among the population of viruses produced on a given day?
c) Number of all possible 1-base mutants is 3 ∗ 104 . Let’s say that one of these
mutations leads to antibiotic resistance. Assuming that all viruses produced on
a given day are different (or as diverse as they can be), what is the probability
that antibiotic resistance will appear on a given day?
d) Repeat b-c with two-base-mutations. Number of all possible two base mutants is approximately 4.5 ∗ 108 .
e) What are your conclusions on development of antibiotic resistance with these
data?
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