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  3. 6 let the random variables x and y represent the...

Question: 6 let the random variables x and y represent the...

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6. Let the random variables X and Y represent the population of two species/organisms that compete with each other for survival. Suppose that the probability density function p(x, y) of these random variables is proportional to rye-(az+By) (a) What would be an appropriate sample space S for the random variables? Justify your choice of S. (b) Let p(x, y)-Krye-(az-+8v). What should be the proportionality constant K? (c) What is the most probable set of populations (X, Y)? (Hint: The probability density function p(x, ) is at its peak for the most probable set of populations) What is the probability of one of the two or both species going extinct? (d) Set up a double integral for the probability of one of the two species dominating the other by more than by a factor of two - i.e.,X 2 2Y or 2X < Y. Sketch the region of integration for the double integral. Do not evaluate the integral.8. Suppose that (X,Y) is the outcome of a random event occuring in some sample space S and the probability distribution is determined by the probability density function p(x, y) (see last homework to review). If g(x, y) is any function that assigns a real number to each point (x, y) in the sample space, then the expected values of X and Y are given by 40 y p(x, y) dA Note these are the same as the equations for the center of mass of a lamina S with density p(x, y).) Find the expected populations X and Y from problem 6 of Homework 1 using α-β-1.

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