# Question about means and variance

• Oct 28th 2012, 12:44 PM
florx
Question about means and variance
Hi, if a problem gave you F(x) = x + 2, x > 0

Would you derive F(x) to get f(x) in order to solve for its mean and variance?

Thank you for your time.
• Oct 28th 2012, 06:47 PM
chiro
Re: Question about means and variance
Hy florx.

What is F(x)? It can't be a probability function (cumulative or not) for a continuous variable since it doesn't have right properties.

Maybe you should explain the entire question and show us what you are trying to do.
• Oct 29th 2012, 12:45 AM
florx
Re: Question about means and variance
Hi chiro, thank you for attempting to help me with this problem. To be specific, the question says, "Computer the mean and variance of random variable X, when Fx(x) = x + 2, x > 0."

Note: There is a subscript x to the right of F in the function Fx(x).
• Oct 29th 2012, 12:56 AM
chiro
Re: Question about means and variance
Is F(x) the cumulative distribution? If so this is not a valid CDF. Also is x discrete or continuous?

Remember that the total probability must be equal to 1. If you have the question available electronically, maybe you could show us that to get some context.
• Oct 29th 2012, 02:12 AM
florx
Re: Question about means and variance
Sorry for confusing you, I am trying to learn how to solve a problem I have by using a simpler question just so I can get the concept. I have now noticed that Fx(x) = x + 2, x > 0 is definitely not going to work since it will equal to a number greater than 1. That is my fault and I apologize.

Fx(x) = 2 - x, for x is between 1 and 2.

Thank you thus far for continuing to help me chiro.
• Oct 29th 2012, 02:54 AM
chiro
Re: Question about means and variance
Is F(x) the cumulative distribution or the probability density function?

I'll assume its the PDF (since the other one can't be right) and integrating across 1 to 2 gives (2*2 - 2^2/2) - (2*(1) - 1^2/2) = 2 - 3/2 = 1/2 so this isn't right either.

If we consider f(x) = x + 2 where x > 0 we can make this work by making our range x^2/2 + 2x = 1 which means x^2 + 4x - 2 = 0 or x = -4 + SQRT(16 - 4*1*-2)/2 or x = -2 + SQRT(6).

If you need to figure out something, you should post the whole question without any omission.
• Oct 29th 2012, 10:31 AM
florx
Re: Question about means and variance
Well I know that we probably need to use these equations:

These equations uses a probability density function. However I am given a probability distribution function and I am trying to find their mean and variance. So I am asking would I differentiate the F(x) given to get the f(x) and then use the two equations above to solve for the mean and variance?

Problem:
F(x) = 1 - e^(-x/10), with x > 0.

Sorry for not being direct with this problem. I just wanted to create a problem of my own so I could understand the concept better to apply it to the problem I have. And yes I believe the function is distribution to answer your question.

----------------------------------------------------
To test, I have derived the F(x) and gotten f(x) = e^(-x/6)/6. However I'm not sure what the upper limit should be if I were to integrate x(e^(-x/6)/6) to get the mean. If I put from 0 to inifinity I would get the answer of 6.
• Oct 29th 2012, 08:29 PM
chiro
Re: Question about means and variance
So this is the cumulative density function so you need to differentiate to get the probability density function.

If you want to make your own density function then the easiest thing to do (if it's not based on any real assumptions or process) is to pick a function and a starting point and solve for the boundary point where the integral between lower and upper limits is equal to 1 (like I did above).

As for your particular problem, if you are stuck just show us how you did the integral and where you got stuck.

If you want to derive distributions that have specific properties, then be aware that this is not quite easy especially if you are learning this for the first time: deriving distributions that have particular attributes in mind (like binomial, chi-square, F-distribution, etc) are things that professional research statisticians do with a lot of experience.
• Oct 29th 2012, 11:42 PM
florx
Re: Question about means and variance
Thank you for your continuous help. I am stuck on finding the limit I must use to differentiate the density function f(x) = x(e^(-x/10)/10) to find out the mean. Is the limit from 0 to infinity? Because the answer would be ten if it is.
• Oct 29th 2012, 11:56 PM
chiro
Re: Question about means and variance
The limits of the density functions (both PDF and CDF) don't change since they are both referring to the same random variable with the same domain.

You have the correct answer of 10 for the mean.
• Oct 30th 2012, 12:08 AM
florx
Re: Question about means and variance
Ah okay. With that the variance should come easy too. Again, thank you so much for helping me with this problem.