increment of Brownian motion squared

Question

$(W_t)_{t \geq 0}$ is Brownian motion,

assume t>s, does $E[(W_t-W_s)^2W_s^2]=(t-s)s$ ?

In other words, are $(W_t-W_s)^2$ and $W_s^2$ independent?

Answer 1

Hint:

1. If two random variables $X$ and $Y$ are independent, then $X^2$ and $Y^2$ are independent.
2. The random variables $W_t-W_s$ and $W_s$ are independent since $(W_t)_{t \geq 0}$ is a Brownian motion.
3. Conclude.