Exercises 1-6 refer to the following systems of equations: (i) dz (i) dz = 10x (1-) - 20xy =0.3x - T0 di di dy Sy+ 20 *= 15y (1-) + 25zy. di di 1. In one of these systems, the prey are very large animals and the predators are very small animals, such as elephants and mosquitoes. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and very small prey. Determine which system is which and provide a justification for your answer. 2. Find all equilibrium points for the two systems. Explain the significance of these points in terms of the predator and prey populations. 3. Suppose that the predators are extinct at time to = 0. For each system, verify that the predators remain extinct for all time.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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