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Calculus for Biologists Lab Math 1180-002 Spring 2012 Assignment #9 - Applications of the Binomial Distribution Report due date: Tuesday, March 27, 2012 at 9 a.m. ? Please number your responses to each problem in your write-up. 1. Describe in detail what each line (and comma-separated element) of the following code does. Be very specific. cumulative = matrix(0,length(time),1) for (j in 1:length(time)){ cumulative[j] = sum(binom(0:M,p(time[j]))) } 2. Include Plot 9.1 from lab 9. (a) Explain the shape of the curves for k = 10 and k = 100. (b) Why are the maximal heights of each of the curves for k = 10, . . . , 90 not larger in this plot? (c) What number of awesomemols is most likely present at 30 seconds? (d) What is the sum over all k of the p.d.f. at 30 seconds? Why does this make sense? n (Recall that the p.d.f. for this problem is [p(t)]k [1 − p(t)]n−k ). k 3. As t → ∞, what would you expect is the most likely number of awesomemols inside the cell? Explain. 4. (a) Evaluate lim Var(number of molecules inside). Use what you know about binomial distributions to t→∞ answer this. (b) Give an explanation for the long-term variance you determined. 5. Include Plot 9.2 from lab 9. (a) What information does this curve give you in terms of the effective threshold? (b) Explain the shape of the curve for (approximately) the following intervals: ? 0 – 3 sec ? 5 – 10 sec ? 15 – 20 sec (c) How does the shape of this curve relate to those in Plot 9.1? 6. Include Plot 9.3 from lab 9. (a) Describe the major differences you notice between this graph and that in Plot 9.2. (b) How often must injections be made to sustain increased AwesomeAox production? (c) Given the frequency of injections, do you think a devised method for systemic release of the awesomemolcontaining solution would be practical? Explain. 7. (a) What change in the awesomemols leaving probability would provide a better systemic treatment method utilizing this experimental solution for an individual with antioxidant deficiency? (b) How would such an alteration affect the cumulative probability curve of Plot 9.2? ? Save your report as a .pdf file and upload it to Blackboard. 1 of 1 L9