You are given the following three observations:

0.74  0.81  0.95

You fit a distribution with the following density function to the data:

[[math]] f(x) = (p+1)x^p, \, 0 \lt x \lt 1, p \gt -1. [[/math]]

Calculate the maximum likelihood estimate of [math]p[/math].

• 4.0
• 4.1
• 4.2
• 4.3
• 4.4

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: D

[[math]] \begin{aligned} L( p ) &= f (0.74) f (0.81) f (0.95) = ( p + 1)0.74 p ( p + 1)0.81 p ( p + 1)0.95 p \\ & = ( p + 1)3 (0.56943) p \\ l( p ) &= \ln L( p) = 3\ln( p + 1) + p \ln(0.56943) \\ &= l^{'}(p) = \frac{3}{p+1} - 0.563119 = 0 \\ &= p + 1 = \frac{3}{0.563119} = 5.32747, p = 4.32747. \end{aligned} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.