Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Let beta be the longer mean. The shorter mean is beta divided by two.

E[XA]=β,E[XB]=β2E[X_A]=\beta,\qquad E[X_B]=\frac{\beta}{2}

Model

Model

Write the supplied survival probability under the exponential model.

e−15/β=0.046656e^{-15/\beta}=0.046656

Compute

Compute

Express the required exponent as two-thirds of the known exponent.

Pr⁡(XB>5)=e−5/(β/2)=e−10/β\Pr(X_B>5)=e^{-5/(\beta/2)}=e^{-10/\beta}
e−10/β=(e−15/β)2/3e^{-10/\beta}=\left(e^{-15/\beta}\right)^{2/3}
Pr⁡(XB>5)=0.0466562/3=0.1296\Pr(X_B>5)=0.046656^{2/3}=0.1296

Answer

Answer

The required survival probability rounds to 0.13.

0.13(D)\boxed{0.13\quad\text{(D)}}