Function of random variables

**roshanhero** · July 28th 2009, 06:02 AM

If X is a random variable with distribution function F(x).Find the distribution of Y=aX+b

**Moo** · July 28th 2009, 08:26 AM

Hello,

In general, F denotes the cumulative density function. But please let us know if it is not.

$F(x)=\mathbb{P}(X\leq x)$

So the cumulative density function of aX+b, let's call it G(x) is :
$G(x)=\mathbb{P}(aX+b\leq x)=\mathbb{P}(aX\leq x-b)=\begin{cases} F\left(\tfrac{x-b}{a}\right) \quad \text{if } a>0 \\ 1-F\left(\tfrac{x-b}{a}\right) \quad \text{if } a<0 \\ \text{depends on b if } a=0 \end{cases}$

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