Show that a gamma pdf has the unique mode r 1 that is, show that the function f Y (y) r (r)y r1 e y takes its maximum value at y mode r1 and at no other point

Question: Show that a gamma pdf has the unique mode r 1/; that is, show that the function f Y (y) = r /

Show that a gamma pdf has the unique mode r − 1/λ; that is, show that the function f_Y(y) = λ^r/ Г(r)y^r−1e^−λy takes its maximum value at y_mode = r−1/λ and at no other point.

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