Question

Let f(x) = 1/pi(1+x^2), where x ∈ R.
a. Show that f is a probability function.

b. Find the mean of the random variable X with this probability function.

(Note: R is all real numbers.)

UPDATE: I've already figured out a), which is 1. However, I still need help on b). Please let me know. Thanks.

Answer 1

for a probability distribution function\[\int_{-\infty}^{\infty} f(x)dx=1\]

Answer 2

can you do this indefinite integral?

Answer 3

I think for a) you can do indefinite, but I don't know about b).

Answer 4

a. so as turing said if
\[\int\limits_{-\infty}^{\infty}f(x)dx=1\]
it is a probability density function and the mean will be
\[\int\limits_{-\infty}^{\infty}xf(x)dx\]

Answer 5

I don't why I said indefinite, I meant "improper"

Answer 6

It should be improper.

Answer 7

it is when the bounds have an infinity in them lol

Answer 8

lol, yeah I had to make sure.

Answer 9

the second integral can be done by parts
any problems you are having specifically?

Answer 10

I think I know what to do with a) now. However, I still don't know what to do for b).

Answer 11

integration by parts\[dv=\frac1{1+x^2}dx\]\[u=x\]

Answer 12

though I don't suppose that will converge... :/

Answer 13

No I don't think it should.