2nd continuous random variable

**ban26ana** · October 1st 2008, 07:18 PM

Consider the continuous random variable X with probability density function as:

Find
a) the value of C
b) the distribution function of X
c) the second moment about the origin of X.

Again, infinity confuses me. I can't figure out how to do the problem because it's not a "normal" number.

**mr fantastic** · October 2nd 2008, 12:21 AM

Originally Posted by ban26ana

Consider the continuous random variable X with probability density function as:

Find
a) the value of C
b) the distribution function of X
c) the second moment about the origin of X.

Again, infinity confuses me. I can't figure out how to do the problem because it's not a "normal" number.

a) $\int_{-\infty}^{+\infty} \frac{C}{1 + x^2} \, dx = \lim_{\alpha \rightarrow \infty} \left[ C \tan^{-1} x\right]_{-\alpha}^{\alpha} = C \pi \, ....$

b) Do you mean the cumulative density function $F(x) = \Pr(X < x)$ .

c) Try to calculate $E(X^2)$ . You'll find it's undefined. The mean (first moment about the origin) of X is also undefined. This is a famous property of this particular pdf (which is an example of a Cauchy distribution: http://en.wikipedia.org/wiki/Cauchy_distribution).

**ban26ana** · October 2nd 2008, 03:11 AM

[quote=mr fantastic;196408]
b) Do you mean the cumulative density function $F(x) = \Pr(X < x)$ >
[quote]
I assume so. Our professor calls it the distribution function.

**mr fantastic** · October 2nd 2008, 04:16 AM

Originally Posted by ban26ana

Originally Posted by mr fantastic

b) Do you mean the cumulative density function $F(x) = \Pr(X < x)$

I assume so. Our professor calls it the distribution function.

$F(x) = \Pr(X < x) = \frac{1}{\pi} \int_{-\infty}^{x} \frac{1}{1 + t^2} \, dt = \frac{1}{\pi} \lim_{\alpha \rightarrow - \infty} \left[ \tan^{-1} t \right]_{\alpha}^{x} = \, ....$

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