# Thread: normal distribution

1. ## normal distribution

need help with this dont know were to start

question

suppose X1 and X2 are identically distributed independent standard normal random variables
$
\mu =0, \sigma =1
$

write down the joint pdf of (X1, X2). determine the pdf of the linear transformation y1 =2x1-x2, y2=x1-2x2.

determine the correct range of this random variable.

any help appreciated

2. You start with the joint density of X1 and X2.
Then you solve for X1 and X2 in terms of the Y's.
You substitute into the first density and you multiply the absolute value of the jacobian.
This is just like the change to polar in calc3.

3. thanks

but how do you solve for x1 and x2.

abit more help please

4. Originally Posted by kermit
thanks

but how do you solve for x1 and x2.

abit more help please
$y_1 = 2x_1 - x_2$ .... (1)

$y_2 = x_1 - 2x_2$ .... (2)

are a pair of linear simultaneous equations and at this level you are well and truly expected to know how to solve such a system. Perhaps use the elimination method ......? Start by multiplying equation (2) by -2 and then adding the two equations together .... ? etc.