Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Write the exponential survival and density functions with an unknown positive rate.

S(x)=e−λxS(x)=e^{-\lambda x}
f(x)=λe−λx,x>0f(x)=\lambda e^{-\lambda x},\qquad x>0

Model

Model

Calibrate the rate from the supplied survival point.

e−4λ=0.30e^{-4\lambda}=0.30
λ=−ln⁡(0.30)4\lambda=-\frac{\ln(0.30)}4

Compute

Compute

Rewrite the exponential factor using the calibrated four-year survival probability.

e−λx=e(ln⁡0.30)x/4=(0.30)x/4e^{-\lambda x}=e^{(\ln 0.30)x/4}=(0.30)^{x/4}
f(x)=−ln⁡(0.30)4(0.30)x/4f(x)=-\frac{\ln(0.30)}4(0.30)^{x/4}

Answer

Answer

This is the density shown in choice E.

−ln⁡(0.30)4(0.30)x/4(E)\boxed{-\frac{\ln(0.30)}4(0.30)^{x/4}\quad\text{(E)}}