University of Alberta Guide/STAT/222/Independence and Conditional Expectations

The basic idea behind independence is that if your random variables (or vectors) are independent then the combination of several of these random variable/vectors $$\left(P\left(X_{1} \leq x_{1}, ..., X_{d} \leq x_{d}\right)\right)$$ can be multiplied together.

Given $$T_{1}, T_{2}\,$$ are independent $$\lambda\,\mbox{-Exponential}$$ random variables, then $$E\left[T_{1} + T_{2}\right] = E\left[2T_{1}\right] = E\left[2T_{2}\right]$$. But $$\mbox{Var}\left(T_{1} + T_{2}\right) < \mbox{Var}\left(2T_{1}\right)$$. This is because $$\mbox{Var}\left(T_{1} + T_{2}\right) = \mbox{Var}\left(T_{1}\right) + \mbox{Var}\left(T_{2}\right) = 2\mbox{Var}\left(T_{1}\right)$$ by independence, whereas $$\mbox{Var}\left(2T_{1}\right) = 2^{2}\cdot\mbox{Var}\left(T_{1}\right)$$.

See the equations section for some more examples.