1. ## Density function, varianceetc.

Hey!
I need to solve these problems, but I really have no idea how to do it.. All help is really welcome!

Thank you!

2. ## Re: Density function, varianceetc.

Originally Posted by missmathematica
Hey!
I need to solve these problems, but I really have no idea how to do it.. All help is really welcome!

Thank you!
Have you looked in your text to find out what properties a density function must have? That would be where to start. It's hard to solve problems about density functions if you don't know what they are. So, first, tell us the properties of a density function.

3. ## Re: Density function, varianceetc.

For Part (1)
$f(x) = c$, where $-1<x<2$ and 0 otherwise,

Notice that since f(x) is the probability function. Its sum over the probability space should be equal to all (Remember "All probabilities sum up to one").
so,

$$\int^{-\infty}_{-\infty}f(x) dx = 1$$

In your case, given the limits,
$$\int^2_{-1}f(x) dx = \int^2_{-1} c dx = 1$$
$$3c=1,$$ so $c=1$

Variance is similarly calculated as:
$$Var(X) = \int_{-1}^{2}x^2f(x)dx - E(X)$$

Part(2)

Expectation is calculated as:
$$E(X) = \int^{\infty}_{-\infty}xf(x) dx$$

$$E(X) = \int^{2}_{-1}xc dx = \frac{5}{3}$$

Part(3)
$$F(x) = P[X<x] = \int^x_{-\infty}f(x)dx$$
$$F(x) = P[X<x] = \int^x_{-1}\frac{1}{3}dx$$
$$F(x) = \frac{1}{3}(x+1)$$

Hope this helps.

4. ## Re: Density function, varianceetc.

Originally Posted by sjaffry
For Part (1)
$f(x) = c$, where $-1<x<2$ and 0 otherwise,
Notice that since f(x) is the probability function. Its sum over the probability space should be equal to all (Remember "All probabilities sum up to one"). so,
$\int^{-\infty}_{-\infty}f(x) dx = 1$ yes

In your case, given the limits,
$$\int^2_{-1}f(x) dx = \int^2_{-1} c dx = 1$$
$$3c=1,$$ so $c=1$ NO!.
Actually $c=\dfrac{1}{3}$

Now redo the other parts for yourself.

5. ## Re: Density function, varianceetc.

Originally Posted by Plato
Actually $c=\dfrac{1}{3}$

Now redo the other parts for yourself.
Yes you are right. It was a typo. Thanks for letting me know

6. ## Re: Density function, varianceetc.

This makes me wonder why bother with this help board. I suggest the OP look in her text for some definitions, which she really needs to do, and others just give a complete solution, modulo typos, before the OP returns.

7. ## Re: Density function, varianceetc.

Originally Posted by Walagaster
This makes me wonder why bother with this help board. I suggest the OP look in her text for some definitions, which she really needs to do, and others just give a complete solution, modulo typos, before the OP returns.
If you think it is bad now then you should been here when Soroban was active. He told me more than once to butt out of his posting. He believed in giving complete polished ready to hand in solutions. His rational was simple: it builds confidence. Soroban was a retired community college instructor that had a completely different idea of student needs. None here now are that bad, believe me.

8. ## Re: Density function, varianceetc.

Originally Posted by Walagaster
This makes me wonder why bother with this help board. I suggest the OP look in her text for some definitions, which she really needs to do, and others just give a complete solution, modulo typos, before the OP returns.
I try not to get too involved with how members help others. I do, however, try to keep an eye on it and PM those who make a habit of just giving out full solutions. Right now we don't have any members who fit that category, though there are some new members I'll probably have to talk to at some point. We tend to work with the Platonic method of instruction here but there are times when this it makes more sense to give the solution, such as a post I recently made on PHF to solve a nasty integral.

If you feel that someone is posting solutions too often please feel free to PM me and I'll take a look.

-Dan

9. ## Re: Density function, varianceetc.

Originally Posted by Walagaster
This makes me wonder why bother with this help board. I suggest the OP look in her text for some definitions, which she really needs to do, and others just give a complete solution, modulo typos, before the OP returns.
Thanks for notifying. I only issued the response in full as I thought it would be helpful for @mismathemarica as when I was learning basic probability, I used to stuck at silly things and wanted some external help which was not there.
So I thought to do my little bit. (Your point is also correct that one shoudl give hint to let the OP learn)