Finding probability density, mean and variance of Y=2X+3

**acevipa** · June 6th 2010, 04:47 AM

Suppose X has a probability density given by:

$f(x)=\left\{\begin{array}{cc} \alpha e^{-\alpha x},&\mbox{for}\ x\geq 0\\0 & \mbox{ otherwise}\end{array}\right.$

1) Find the probability density of $Y=2X+3$

2) Find the mean and variance of $Y=2X+3$

Could someone please explain or refer me to someone website which explains the 'scale effects'. I tried reading it in my textbook, but don't really understand it.

**SpringFan25** · June 6th 2010, 04:49 AM

by scale effects do you mean you want an explanation of why

$Var(aX+b) = a^2 Var(X)$
and
$E(aX+b) = aE(x) + b$

?

**acevipa** · June 6th 2010, 04:51 AM

Originally Posted by SpringFan25

by scale effects do you mean you want an explanation of why

$Var(aX+b) = a^2 Var(X)$
and
$E(aX+b) = aE(x) + b$

?

Yeh kind of. I'll give you an example

Say X is a continuous random variable with PDF $f(x)$ and CDF $F(x)$ . Suppose $Y=aX+b$ where $a \ne 0$ has PDF $g(x)$ and CDF $G(x)$ .

What I don't understand is the meaning of $F(x)=G(ax+b)$ and $g(y)=\frac{1}{a}f\left(\frac{y-b}{a}\right)$ for $a>0$

**SpringFan25** · June 6th 2010, 05:14 AM

Ok, lets examine the CDF of Y.

Step 1:
$G(Y) = P(Y<y) = P(aX + b < y)$
$G(Y) = P(Y<y) = P(X < \frac{y - b}{a})$
$G(Y) = P(X < \frac{y - b}{a}) = F(\frac{y-b}{a})$

Step 2:
Now, to get any pdf from a cdf, we just differenciate.
$g(y) = \frac{dG}{dy} = \frac{dF \left( \frac{y-b}{a} \right)}{dy}$

You can do this derivative easily using the chain rule

$g(y) = \frac{dF \left( \frac{y-b}{a} \right)}{dy} = \frac{1}{a} \times F' \left( \frac{y-b}{a} \right)$
Notation point: $F' \left( \frac{y-b}{a} \right)$ means the derivative of F with respect to x, evaluated at the point $\left( \frac{y-b}{a} \right)$

But by definition, the derivative of F is the pdf (ie, F' = f).
$g(y) = \frac{1}{a} \times f \left( \frac{y-b}{a} \right)$

To explain why F(X) = G(aX + b) you can follow the argument in step 1 again, but try to find F() instead of G(). I've done it as a spoiler in case you get stuck

Spoiler:

**acevipa** · June 6th 2010, 05:43 AM

Originally Posted by SpringFan25

Ok, lets examine the CDF of Y.

Step 1:
$G(Y) = P(Y<y) = P(aX + b < y)$
$G(Y) = P(Y<y) = P(X < \frac{y - b}{a})$
$G(Y) = P(X < \frac{y - b}{a}) = F(\frac{y-b}{a})$

Step 2:
Now, to get any pdf from a cdf, we just differenciate.
$g(y) = \frac{dG}{dy} = \frac{dF \left( \frac{y-b}{a} \right)}{dy}$

You can do this derivative easily using the chain rule

$g(y) = \frac{dF \left( \frac{y-b}{a} \right)}{dy} = \frac{1}{a} \times F' \left( \frac{y-b}{a} \right)$
Notation point: $F' \left( \frac{y-b}{a} \right)$ means the derivative of F with respect to x, evaluated at the point $\left( \frac{y-b}{a} \right)$

But by definition, the derivative of F is the pdf (ie, F' = f).
$g(y) = \frac{1}{a} \times f \left( \frac{y-b}{a} \right)$

To explain why F(X) = G(aX + b) you can follow the argument in step 1 again, but try to find F() instead of G(). I've done it as a spoiler in case you get stuck

Spoiler:

Thanks, I kind of understand that now. Although how do you approach the question. Do you just remember the formula??

**SpringFan25** · June 6th 2010, 05:46 AM

I never saw that formula before actually

The principle i remembered was "Find the CDF of Y and differenciate to get the PDF"

**acevipa** · June 6th 2010, 05:58 AM

Originally Posted by SpringFan25

I never saw that formula before actually

The principle i remembered was "Find the CDF of Y and differenciate to get the PDF"

In my case, do I integrate first to find F(x) then use that to find G(Y) then F(Y) then f(y)??

**SpringFan25** · June 6th 2010, 06:05 AM

Im not clear on what you mean, there are 2 ways to get the pdf of Y.

Short way:
just plug the numbers into the formula we just derived

Long way:
Step 1:
Find the CDF of X

HintYou can either integrate, or notice that we are dealing with the pdf of an exponential distribution. You can lookup the CDF of an exponential distribution

Step 2:
Find the CDF of Y (use the same reasoning i did in the derivation above)

Step 3:
Differentiate the CDF of Y to get the pdf of Y.

**acevipa** · June 6th 2010, 05:43 PM

Originally Posted by SpringFan25

Im not clear on what you mean, there are 2 ways to get the pdf of Y.

Short way:
just plug the numbers into the formula we just derived

Long way:
Step 1:
Find the CDF of X

HintYou can either integrate, or notice that we are dealing with the pdf of an exponential distribution. You can lookup the CDF of an exponential distribution

Step 2:
Find the CDF of Y (use the same reasoning i did in the derivation above)

Step 3:
Differentiate the CDF of Y to get the pdf of Y.

Thanks for that

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