The lifetime of a light bulb has density function, [math]f[/math], where [math]f(x)[/math] is proportional to

[[math]] \frac{x^2}{1+x^3}, \, 0 \lt x \lt5, \, \textrm{and}, \, 0 \, \textrm{otherwise}. [[/math]]

Calculate the mode of this distribution.

• 0.00
• 0.79
• 1.26
• 4.42
• 5.00

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

Solution: C

It is not necessary to determine the constant of proportionality. Let it be c. To determine the mode, set the derivative of the density function equal to zero and solve.

[[math]] \begin{align*} 0 &= f^{'}(x) = \frac{d}{dx} cx^2(1+x^3)^{-1} = 2cx(1+x^3)^{-1} + cx^2[-(1+x^3)^{-2}]3x^2 \\ &= 2cx(1+x^3)-3cx^4 \\ &= 2cx + 2cx^4 -3cx^4 = 2cx - cx^4 \\ &= 2- x^3 \Rightarrow x = 2^{1/3} = 1.26. \end{align*} [[/math]]

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