Find the mode of the
a. Beta distribution
b. Gamma distribution
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Hint: you want to maximize xα−1e−βxxα−1e−βxover x∈(0,∞)x∈(0,∞).
The derivative is e−βx[(α−1)xα−2−βxα−1]=xα−2e−βx(α−1−βx)e−βx[(α−1)xα−2−βxα−1]=xα−2e−βx(α−1−βx), which is zero when x=α−1βx=α−1β or x=0x=0.
If α≥1α≥1, direct inspection shows that x=0x=0 is not the mode, since the pdf is zero there. Thus the other critical point α−1βα−1β must be the mode.
If α<1α<1 the pdf has a positive asymptote at x=0x=0. Moreover the derivative is strictly negative for all x>0x>0, so it decreases from ∞∞ to 00as x
The derivative is e−βx[(α−1)xα−2−βxα−1]=xα−2e−βx(α−1−βx)e−βx[(α−1)xα−2−βxα−1]=xα−2e−βx(α−1−βx), which is zero when x=α−1βx=α−1β or x=0x=0.
If α≥1α≥1, direct inspection shows that x=0x=0 is not the mode, since the pdf is zero there. Thus the other critical point α−1βα−1β must be the mode.
If α<1α<1 the pdf has a positive asymptote at x=0x=0. Moreover the derivative is strictly negative for all x>0x>0, so it decreases from ∞∞ to 00as x
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