================================================================================ Expectation value: average of random variable ================================================================================ Sample data's average: $$$\bar{x}$$$ Sample data's variance: $$$s^2$$$ which are not "population's characteristics" but they're prediction for characteristics of population. Population's average: $$$\mu$$$ Population's variance: $$$\sigma^2$$$ ================================================================================ $$$\bar{x} \\ =\frac{1}{n}\sum\limits_{i=1}^n x_i \\ =\frac{1}{n}\sum\limits_{\text{all x}} n_x x \\ =\sum\limits_{\text{all x}} x \frac{n_x}{x}$$$ $$$\frac{n_x}{n}$$$ is relative frequency ================================================================================ $$$E[X]=\mu=\sum\limits_{\text{all x}} xp(x)$$$ Best proper value you can expect from random variable X ================================================================================ In the case of where you use continuous random variable $$$E[X]=\mu=\int_{-\infty}^{\infty} xf_x(x)dx$$$ ================================================================================ Variance of discrete random variable * Variance of sample $$$s^2 \\ = \frac{1}{n-1} \sum\limits_{i=1}^n (x_i-\bar{x})^2 \\ = \sum\limits_{\text{all x}} (x-\bar{x})^2 \frac{n_x}{n-1}$$$ * $$$\frac{n_x}{n-1}$$$: relative frequency * Variance of population $$$\sigma^2 = \sum\limits_{\text{all x}} (x-\mu)^2 p(x)$$$ ================================================================================ Variance of continuous random variable * Variance of sample $$$Var[X] \\ = E[(X-E[X])^2] \\ = \int_{-\infty}^{\infty}(x-\mu)^2 f_X(x)dx$$$ * $$$\frac{n_x}{n-1}$$$: relative frequency * Variance of population $$$STD[X] = Var[X]^{\frac{1}{2}}$$$ ================================================================================