Modeling Random Processes in Natural Phenomena: Case Studies and Applications

[vampire2008]

Homework Statement

1. Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two earthquakes before the next volcanic eruption?

2. A certain geiger counter (an instrument that detects individual radioactive decay events) has not been well maintained, and hence after each one it detects, it will not detect another (no matter how many occur) until it has gone a full two seconds with no decays. (for example, if it detects one event at 3 seconds, and this is followed by events at 3.5, 4.5, 6 and 9 seconds, the counter will not detect those at 3.5, 4.5 and 6 seconds.) Assume decay events occur according to a Poisson process with rate 0.5. Find the expected value of the time between events detected by the counter.

3. Briefly explain why it would or would not be a good idea to model the births of humans on Earth over the next year as a Poisson process.

4. A poor soul who has the title of Core Math Director receives emails at exponentially distributed time intervals, with rate 1 per minute. the times required to respond to these are independent, exponentially distributed with rate 2 per minute. However, when the list of unanswered messages gets above 10, his stress level rises and his efficiency goes down, reducing the rate to 1 per minute. Set up a birth/death process that models the email queue for this beleaguered public servant. Specifically, describe the states, and list the birth rate λi and the death rate μi in each state.

5.Customers arrive at a certain restaurant according to a Poisson process with rate 3 per hour. a small percentage of the customers are actually undercover health inspectors (as well as being customers). they come to the restaurant according to a Poisson process with rate 0.01 per hour. if no customer has entered for 1/3 of an hour, what is the probability that the next customer is a health inspector?

 

[lanedance]

hey vampire2008 - so what have you tried?

to get you started on the first one, start with the exponential probability distribution
p(t) = \lambda e^{-\lambda t}

then find the probability distribution p(t)dt for 2 eruptions with the 2nd occurring at t
then assume volcanoes and earthquakes are independent and take it from there

 

[vampire2008]

I tried most of them, but I can't handle them. for problem 1, I don't understand what u told me, can u offer more detail? thank u

 

[lanedance]

show me what you tried?

for problem 1)
the probability of an eruption ocurring at time t (years) is
p(t) = \frac{1}{2}e^{- \frac{1}{2} \lambda t}

the probability of an earthquake ocurring at time t (years) is
p(t) = e^{- t}

now say an earthquake occurs at t=t0, what is the probability of a 2nd earthquake occurring at t = t0+t1?

if you can solve that the solution to the complete problem should become obvious

 

[vampire2008]

since T1,T2~exp(1) and t1~exp(2), thus the desired probability should be P(T1+T2<t1) then use the formula to solve, where T1, T2 are elapsed time of fist earthquake and the elapsed time between the first and second earthquakes respectively, t1 denotes the elapsed time of the first volcanic eruption. is that right? thank u !

 

[lanedance]

not quite

so we want to find the probability of 2 earthquake at time z, let's call it p_2(z)dz

the probability of the first earthquake occurring at time t=u is
p(u)du = e^{- u}dunow assuming the first has aoccurred, the

now assuming the first has occurred, the probability of a 2nd earthquake occurring v years after the first is then
p(v)dv = e^{- v}dv


then probability distribution for 2 earthquake occurring at z=u+v is gievn by
p(z) = \int\int du dv p(v)p(u) \delta (z-u-v)

 

[vampire2008]

I don't understand why there are two volcanic eruption? should be two earthquakes, right?

 

[lanedance]

correct, i have updated above

 

[vampire2008]

thank you, I am thinking about it