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Transformation of Random Variable

hemanth

New member
May 31, 2012
9
If X is a random variable distributed uniformly in [0, Y], where Y is geometric with mean alpha.
i) Is this definition valid for uniform distribution ?
ii) If it is valid, what is the pdf of the transformation Y-X?
 

chisigma

Well-known member
Feb 13, 2012
1,704
If X is a random variable distributed uniformly in [0, Y], where Y is geometric with mean alpha.
i) Is this definition valid for uniform distribution ?
ii) If it is valid, what is the pdf of the transformation Y-X?
If Y is geomtrically distributed with parameter p, that means that...


$$P \{ Y = n\} = p\ (1-p)^{n}\ (1)$$

... and...

$$E \{Y\} = \sum_{n=0}^{\infty} n\ p\ (1-p)^{n} = \frac{1-p}{p}\ (2)$$

If X is uniformely distributed in [0,Y], then is...

$$ P \{Y < x \} = p + p\ \sum_{n=1}^{\infty} \varphi_{n} (x)\ (1-p)^{n}\ (3)$$

... where...

$$\varphi_{n} (x)=\begin{cases}\ 0 & \text{if}\ x < 0 \\ \frac{x}{n} &\text{if}\ 0 \le x \le n \\ 1 &\text{if} x> n \end{cases}\ (4)
$$

Of course the p.d.f. of Y is the derivative of (3). I don't know if all that answers the question i) ...


Kind regards


$\chi$ $\sigma$