# Variance of Continous Random Variable

I have the continous random variable Y, defined such that:

$$Y=3X+2$$
and the PDF of x is zero everywhere but:
$$f(x)=\frac{1}{4}e^{\frac{-x}{4}}, x>0$$

I correctly got the mean like so:
$$\mu=E(h(x))=\int^{\infty}_{0} h(x)f(x)$$
and evaluated it to be 10.

I am unsure as to how I go about getting the variance now. In previous problems I use the usual formula of:

$$\sigma=\int^{b}_{a} (x-\mu)^{2}f(x)$$

or the shortcut of:
$$\sigma^{2}=E[X^{2}]-\mu^{2}$$

However I'm unsure what to do in the case where the random variable is dependent on another with a given PDF function for continuous random variables.

Start with substiting in the expression for Y into $Var(Y)$

Seperate out $Var(X)$

Start with substiting in the expression for Y into $Var(Y)$

Seperate out $Var(X)$

I'm not quite sure how to put that together. To start, I have:
$$Var(Y)=\int^{\infty}_{0} (y-10)^2*f(y)$$

I would start with $Var(Y)=Var(3X+2)=Var(3X)+Var(2)$

I'm not quite sure how to put that together. To start, I have:
$$Var(Y)=\int^{\infty}_{0} (y-10)^2*f(y)$$

If you want to use this form then $\mu_Y=E(Y)=E(3X+2)=3E(X)+2$

If you want to use this form then $\mu_Y=E(Y)=E(3X+2)=3E(X)+2$

I see. So all I need to do is then find the expected value of the function defining the PDF of x?

Ray Vickson
Homework Helper
Dearly Missed
I have the continous random variable Y, defined such that:

$$Y=3X+2$$
and the PDF of x is zero everywhere but:
$$f(x)=\frac{1}{4}e^{\frac{-x}{4}}, x>0$$

I correctly got the mean like so:
$$\mu=E(h(x))=\int^{\infty}_{0} h(x)f(x)$$
and evaluated it to be 10.

I am unsure as to how I go about getting the variance now. In previous problems I use the usual formula of:

$$\sigma=\int^{b}_{a} (x-\mu)^{2}f(x)$$

or the shortcut of:
$$\sigma^{2}=E[X^{2}]-\mu^{2}$$

However I'm unsure what to do in the case where the random variable is dependent on another with a given PDF function for continuous random variables.

Neither X nor Y has mean 10, so you did something wrong. What is the mean of X? How is the mean of Y related to the mean of X? What is Var(X)? How is Var(Y) related to it?

RGV

Neither X nor Y has mean 10, so you did something wrong. What is the mean of X? How is the mean of Y related to the mean of X? What is Var(X)? How is Var(Y) related to it?

RGV

10 is what I got, and it matches the answer in the textbook...

jbunniii
Homework Helper
Gold Member
10 is what I got, and it matches the answer in the textbook...
I calculated E[X] = 4. Please show how you got 10.

Ray Vickson
Homework Helper
Dearly Missed
10 is what I got, and it matches the answer in the textbook...

Well, EX (the mean of X) is not 10, and EY (the mean of Y) is not 10. However, E(3*X - 2) = 10, so maybe you just typed the question incorrectly, and maybe you really meant EY when you seemed to be talking about EX.

RGV

I calculated E[X] = 4. Please show how you got 10.

Here is the integral I solved in order to get the mean:

$$\mu=\int^{\infty}_{0} \frac{1}{4}e^{\frac{-x}{4}}(3x-2)dx=10$$

from the identity where:

$$E[h(x)]=\int^{\infty}_{-\infty} h(x)f(x) dx$$

Anyhow, it turns out that in solving the variance I made an arithmetical mistake, and I did manage to get the correct answer. Thanks for the help!

Ray Vickson
Homework Helper
Dearly Missed
Here is the integral I solved in order to get the mean:

$$\mu=\int^{\infty}_{0} \frac{1}{4}e^{\frac{-x}{4}}(3x-2)dx=10$$

from the identity where:

$$E[h(x)]=\int^{\infty}_{-\infty} h(x)f(x) dx$$

Anyhow, it turns out that in solving the variance I made an arithmetical mistake, and I did manage to get the correct answer. Thanks for the help!

OK, so as I suspected, you computed EY = E(3X-2), but your question originally said Y = 3X+2!

BTW: if 'a' and 'b' are constants, Var(AX+b) = a^2 Var(X) [that is, the constant term 'b' drops out and the factor 'a' gets squared]. Since Var(X) = 4^2 = 16, we have Var(Y) = 3^2 * 16 = 144. This is a lot easier than computing
$$\int_0^{\infty} (y-10)^2 f_Y(y) \, dy.$$

RGV