# basic weiner process

• Sep 23rd 2007, 02:08 AM
BruceBronson
basic weiner process
Can someone help me with this, i need to find the distribution of

W_0 + W_2 - W_3 + 2W_4 where W is the standard weiner process

Any help would be greatly appreciated thanks.
• Sep 23rd 2007, 06:04 AM
CaptainBlack
Quote:

Originally Posted by BruceBronson
Can someone help me with this, i need to find the distribution of

W_0 + W_2 - W_3 + 2W_4 where W is the standard weiner process

Any help would be greatly appreciated thanks.

Look at the definition of a Wiener Process then:

Group the terms together into non-overlapping time pairs:

\$\displaystyle
W_0 + W_2 - W_3 + 2W_4 = W_0+W_2+W_3 + 2(-W_3+W_4)
\$

......... \$\displaystyle
= W_0+2W_2+(-W_2+W_3) + 2(-W_3+W_4)
\$

......... \$\displaystyle
= 3W_0+2(-W_0+W_2)+(-W_2+W_3) + 2(-W_3+W_4)
\$

Then the three bracketed terms are zero mean Guassian distributed and independent,
with variances 4, 1 and 2. So the required distribution is Gaussian with mean \$\displaystyle 3W_0=0\$ and
variance \$\displaystyle 7\$

RonL