MA2102

Probability and Statistics

Lecture-13

Mathematical Expectation

Definition:

• Let $X$ be a discrete random variable with $PMF$ $p_X$ and support $S_X$, and if $\sum_{x\in S_X}|x|p_X(x)<\infty$, then we say that expectation of $X$ exists, defined as $\sum_{x\in S_X}xp_X(x)$ (i.e Expectation exists if $\sum_{x\in S_x}xp_X(x)$ converges absolutely)

• Let $X$ be a continuous random variable with $PDF$ $f_X(x)$ and if $\int_{-\infty}^{\infty}|x|f_X(x)dx<\infty$, then we say expectation exists, and defined as $\int_{-\infty}^{\infty}xf_X(x)dx$ (i.e Expectation exists if $\int_{-\infty}^{\infty}xf_X(x)dx$ (as improper integral)converges absolutely)