Finding the mean and variance of random variables (discrete and continuous, specifically of indicators) and their properties.

Question

I was just wondering if someone can help me understand the steps required to finding the mean and variance of random variables (discrete and continuous, specifically of indicators) and their properties. I know that the mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Also the Variance of a discrete random variable X measure the spread or variability of the distribution.

Answer 1

Let $X$ be a random variable with distribution $P(X)$. If $P(X)$ has an associated density $f(X)$, the mean is given by: $$\langle X\rangle=\int_{-\infty}^{+\infty}xf(x)\,dx$$

For a discrete distribution you can use: $$\langle X\rangle=\sum_{k=0}^{+\infty}kP(k)$$

The variance is defined by the expression $$\mathrm{Var}(X):=\langle (X-\langle X\rangle)^2\rangle,$$ where $\langle X \rangle$ is the mean as defined above.