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Math 4600, Homework 8 1. (Computing) In class, we derived the survivorship function, S(t), of the exponential distribution with constant event rate; recall that S(t) was the probability that an event (such as dying, or a virus leaving the cell) has not yet occur at time t, given that the event occurs at rate λ. In a large population, it can also be interpreted as the proportion of individuals for which the event has not yet occurred. The goal of this question is for you to use the derivation of S(t) in class to extend this to the case where the event rate is not constant. a. Suppose that the probabilistic death rate of a population of bacteria is changing with time: let λ(t) denote the death rate at time t. Derive a general form of the survivorship function, S(t), for λ(t). Assume that the probability of being alive at time 0 is 1. b. Different groups of bacteria have different death rates: suppose group I has a rate of λ(t) = δt, group δ . Calculate the specific S(t) for II has a rate λ(t) = δ, and group III has a death rate of λ(t) = α+t each group. Make sure S(0) = 1. c. Write a sentence describing each of these death rates; how do they differ qualitatively in describing when organisms die? d. Simulate 1000 bacteria from each group, and plot the survivorship curves (fraction alive at time t) for each group on the same graph. Choose δ = 2, α = .25. Add on a plot of your function S(t) that you derived (it should match your simulation). e. Recall the cdf is related to the survivorship function by F (t) = 1 − S(t), and the pdf satisfies R∞ f (t) = F 0 (t). Use the fact that the mean lifespan TĚ„ = t f (t) dt to find the mean lifespan of bacterial 0 types II and III. Show the derivation, even if you know the answer. Show a comparison of these results with means obtained from your simulation. 2. Consider a process of molecules leaving a cell, independent of each other, with a constant probabilistic rate of 2 per second. a. Find the 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? k −λt is the 3. Verify that the Poisson distribution probability model over time period t, P r(N = k) = (λt)k!e k solution to the Master Equations we derived in class dP = λP − λP . Recall P (t) is the probability k−1 k k dt that k events have occurred by time t, k ∈ {0, 1, 2, ...}. 1