Question:Means and variances of linear combinations of random variables

**demon** · November 7th 2006, 07:34 AM

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?

**Plato** · November 7th 2006, 07:53 AM

$\begin{array}{l}<br /> E(Y) = \int\limits_0^\infty {Yf(x)dx} = \int\limits_0^\infty {\left( {3x - 2} \right)f(x)dx} \\ <br /> E(Y^2 ) = \int\limits_0^\infty {Y^2 f(x)dx} = \int\limits_0^\infty {\left( {3x - 2} \right)^2 f(x)dx} \\ <br /> V(Y) = E(Y^2 ) - E^2 (Y) \\ <br /> \end{array}$

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