Review of Terminology and Concepts

• Random variable: A variable that assumes values in a irregular pattern (or no particular pattern).
• Discrete random variables: Let X be a random variable. If the number of possible values of X is finite, or countably infinite, X is called a discrete random variable.

  Let $x_i$ be all possible values of X, and $p(x_i) = P (X = x_i)$ be the probability that $X = x_i$, then $p(x_i)$ must meet the following conditions.

  1. $p(x_i) \ge 0$ for all $i$.
  2. $\sum_{i=1}^\infty p(x_i) = 1.$

  Examples: example 6.1 and 6.2 on p. 186

• Contineous random variables: If the values of X is an interval or a collection of intervals, then X is called a contineous random variable.

  For contineous random variable, its probability is represented as
  

  $$P(a \le X \le b) = \int_a^b f(x) dx$$
  
  
  The function $f(x)$ is called the probability density function (pdf) of the random variable $X$, which has to meet the following condition.
  1. $f(x) \ge 0$ for all $x$.
  2. $\int f(x) dx = 1$.
  3. $f(x) = 0$ if $x$ is not in $R_X$.
  Example 6.3 on page 187.

• Cumulative distribution function. The cumulative distribution function (cdf), denoted by $F(x)$, measures the probability that the random variable $X$ assumes a value less than or equal to $x$, $F(x) = P(X \le x)$.
  • If $X$ is discrete, then $F(x) = \sum_{{\rm all}~x_i \le x} p(x_i)$
  • If $X$ is contineous, the $F(x) = \int_{-\infty}^x f(t) dt$

  Some propertities of cdf include:

  1. $F$ is a non-decreasing function. If $a < b$ then $F(a) \le F(b)$.
  2. $lim_{x \rightarrow \infty} F(x) = 1$.
  3. $lim_{x \rightarrow -\infty} F(x) = 0$.

  Example: 6.4, 6.5 on page 189.

• Expectation and variance. Expectation essentially is the expected value of a random variable. Variance is a measure how a random variable varies from its expected value.
  • For discrete random variables
    
    $$E(X) = \sum_{{\rm all}~i} x_i p(x_i)$$
    
    

  • For continueous random variables
    
    $$E(X) = \int_{-\infty}^{\infty} xf(x)dx$$
    
    

  • For discrete or continueous random variables, its variance is
    
    $$V(X) = E[(X - E[X])^2]$$
    
    
    which has an identity
    
    $$V(X) = E(X^2) - [E(X)]^2$$
    
    

  • A more frequently used practical measure is standard deviation of a random variable, which is expressed as the same units as that of expectation.
    
    $$\sigma = \sqrt{V(X)}$$
    
    

  Examples: 6.6, 6.7 on page 191.

• The mode. The mode is used to describe most frequently occured values in discreate random variable, or the maximum value of a continueous random variable.