1. ## Brownian 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.

2. ## Re: 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.

3. ## Re: 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????

4. ## Re: Brownian Motion

Hint: If two variables are independent then E[XY] = E[X]E[Y]