Consider a more complex PDF, given by: m > 12, > 0, and () is the gamma function

Question

Consider a more complex PDF, given by:

m > 12, > 0, and () is the gamma function (given as GAMMA in Maple).

Use Maple to do the following:

1. [3marks] Derive the mean (not sample mean) analytically for f(x). HINT: Use “assume” in Maple to enforce the constraints on m and in the PDF.

2. [2 marks] Derive the variance analytically for f(x).

3. [4marks]On the same graph,plot the PDF for =1 and m=0.5,1,2,and 4. Label
the graphs appropriately.

4. [4 marks] Derive an expression for the x value at which f(x) attains its maximum value (the highest point of f(x)) in terms of m and . HINT: Use concepts from calculus to find the value x* at which the function attains its maximum/minimum value of a function, and show that f(x*) is indeed the maximum (and not the minimum).

5. [4 marks] Using the result in (4), compute the maximum value of f(x) for:
m = 1, = 1
m = 3, = 0.5

Explanation / Answer

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