Hi!!! I hope someone can help me with the following exercise...

n>=1, 0=t_{0}<t_{1}<...<t_{n}, a_{1},a_{2},...,a_{n} ε R. Show that the random variable a_{1}*B(t_{1})+...+a_{n}*B(t_{n}) is normally distributed and find its mean value and variance.

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- Apr 28th 2013, 01:45 PMmathmariBrownian Motion
Hi!!! I hope someone can help me with the following exercise...

n>=1, 0=t_{0}<t_{1}<...<t_{n}, a_{1},a_{2},...,a_{n} ε R. Show that the random variable a_{1}*B(t_{1})+...+a_{n}*B(t_{n}) is normally distributed and find its mean value and variance. - Apr 28th 2013, 11:52 PMchiroRe: Brownian Motion
Hey mathman.

Hint: Use the property of a Brownian motion that if if B_a(t) and B_b(t) are brownian motions over disjoint times then they are both independent. - Apr 29th 2013, 02:35 PMmathmariRe: Brownian Motion
How can I find the mean value E(a_{1}*B(t_{1})+...+a_{n}*B(t_{n})) and the variance Var(a_{1}*B(t_{1})+...+a_{n}*B(t_{n}))? Using the property of a Brownian motion that E(B(t)-B(s))=0 and Var(B(t)-B(s))=t-s, 0<=s<t????

- Apr 29th 2013, 07:22 PMchiroRe: Brownian Motion
Hint: If two variables are independent then E[XY] = E[X]E[Y]