Show the following. (Geometric)

**universalsandbox** · January 27th 2009, 06:25 AM

Consider that if X and Y are identically independent and geometrically(p) distributed, show the following:

P( X ≥ Y ) = 1 / ( 2 - p )

**mr fantastic** · January 27th 2009, 06:48 PM

Originally Posted by universalsandbox

Consider that if X and Y are identically independent and geometrically(p) distributed, show the following:

P( X ≥ Y ) = 1 / ( 2 - p )

$\Pr(X \geq Y) + \Pr(X \leq Y) - \Pr(X = Y) = 1$ .

By symmetry, $\Pr(X \geq Y) = \Pr(X \leq Y)$ .

Therefore $2 \Pr(X \geq Y) = 1 + \Pr(X = Y)$ .

$\Pr(X = Y) = \Pr(X = 0, Y = 0) + \Pr(X = 1, Y = 1) + \Pr(X = 2, Y = 2) + \, ....$

$= p^2 + (1 - p)^2 p^2 + (1 - p)^4 p^2 + \, ....$

(since X and Y are independent)

$= p^2 (1 + (1 - p)^2 + (1 - p)^4 + \, .... ) = p^2 \cdot \frac{1}{1 - (1 - p)^2} = \frac{p}{2 - p}$ .

Therefore $2 \Pr(X \geq Y) = 1 + \frac{p}{2 - p}$

etc.

Thread: Show the following. (Geometric)

LinkBack

Thread Tools

Display

Show the following. (Geometric)

Similar Math Help Forum Discussions

Geometric proof that the geometric mean is less than or equal to the arithmetic mean.

Show that the inequality is true | Geometric Mean

Geometric Progression or Geometric Series

how to show show this proof using MAX

Arithmetic sequence, geometric sequence, & geometric series

Search Tags