# Question:Means and variances of linear combinations of random variables

• November 7th 2006, 07:34 AM
demon
Question:Means and variances of linear combinations of random variables
The length of time, in minutes, for and airplane to obtain clearance for take off at a certain airport is a random variable Y=3X-2, where x has the density function

f(x) = { 1/4e^(-x/4), x>0
0, elsewhere,

Find the mean and variance of the random variable Y.

the answer from the text book are: mean of Y = 10, variance = 144;

anyone know how to get the answers?
• November 7th 2006, 07:53 AM
Plato
$\begin{array}{l}
E(Y) = \int\limits_0^\infty {Yf(x)dx} = \int\limits_0^\infty {\left( {3x - 2} \right)f(x)dx} \\
E(Y^2 ) = \int\limits_0^\infty {Y^2 f(x)dx} = \int\limits_0^\infty {\left( {3x - 2} \right)^2 f(x)dx} \\
V(Y) = E(Y^2 ) - E^2 (Y) \\
\end{array}$