Question

10. The length of time, in minutes, for an airplane to wait for clearance to take off at a certain

airport is a random variable Y = 3X − 2, where X has the density function

f(x) = {1/4^e−x/4, 0.

x > 0

elsewhere,

Find the mean and variance of the random variable Y.

Answer 1

We have $X\sim Exp\left( \frac{1}{4} \right)$ , from which $EX=\frac{1}{1/4}=4,DX=\frac{1}{\left( 1/4 \right) ^2}=16$

Then

$EY=E\left( 3X-2 \right) =3EX-2=3\cdot 4-2=10\\DY=D\left( 3X-2 \right) =9DX=9\cdot 16=144$