Let a, 3 > 0 be two fixed real numbers. Define the function: Bas 2541, *...

 

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Let a, 3 > 0 be two fixed real numbers. Define the function: Bas 2541, * (, 0) f(x; a, 3): = 0. otherwise. The distribution of the pdf f(r; a, 3) is known as the Pareto Distribution, it shows up in many applications, especially in modelling distribution of wealth. You can use R to plot f(r; a, 3) for different values of a and 8. (a) Verify that f(r) is a probability density function. (b) Calculate the expected value of X. What condition do we need to assume on 3 so that E(X) is a finite number? (c) Calculate the variance of X. What condition do we need to assume on 3 so that V(X) is a finite number? Let a, 3 > 0 be two fixed real numbers. Define the function: Bas 2541, * (, 0) f(x; a, 3): = 0. otherwise. The distribution of the pdf f(r; a, 3) is known as the Pareto Distribution, it shows up in many applications, especially in modelling distribution of wealth. You can use R to plot f(r; a, 3) for different values of a and 8. (a) Verify that f(r) is a probability density function. (b) Calculate the expected value of X. What condition do we need to assume on 3 so that E(X) is a finite number? (c) Calculate the variance of X. What condition do we need to assume on 3 so that V(X) is a finite number?