Probability Distribution

1. Random Variables

• Definition

  A random variable is a real-valued function defined on the sample space of a random experiment.

  The domain of a random variable is the sample space of a random experiment, while its co-domain is the set of real numbers.

  Thus, it is denoted as $X:S \rightarrow R$.

2. Discrete Random Variables

• Probability Mass Function (p.m.f.)

  Let the possible values of a discrete random variable $X$ be denoted by $x_1, x_2, x_3, \ldots$ and the corresponding probabilities be $p_i = P[X=x_i]$.

  The function is a p.m.f. if it satisfies the following conditions:

  1. $p_i \ge 0$, for $i=1, 2, \ldots, n$.
  2. $\sum_{i=1}^{n} p_i = 1$.

  Tabular Form:

  $x_i$	$x_1$	$x_2$	$x_3$	$\dots$
  $P[X=x_i]$	$p_1$	$p_2$	$p_3$	$\dots$

• Cumulative Distribution Function (c.d.f.)

  The c.d.f. of a discrete random variable $X$, denoted by $F(x)$, is defined as:

  $$ F(x) = P[X \le x] = \sum_{x_i \le x} P[X=x_i] = \sum_{x_i \le x} p_i $$

• Expected Value (Mean)

  The expected value or arithmetic mean of $X$, denoted by $E(X)$ or $\mu$, is defined by:

  $$ E(X) = \mu = \sum_{i=1}^{n} x_i p_i = x_1 p_1 + x_2 p_2 + \ldots + x_n p_n $$

• Variance and Standard Deviation

  Variance: The variance of $X$, denoted by $Var(X)$ or $\sigma_x^2$, is defined as:

  $$ Var(X) = \sigma_x^2 = \sum_{i=1}^{n} (x_i - \mu)^2 p_i $$

  Simplified computational form:

  $$ Var(X) = E(X^2) - [E(X)]^2 $$

  where $E(X^2) = \sum_{i=1}^{n} x_i^2 p_i$.

  Standard Deviation: The non-negative square root of the variance.

  $$ \sigma_x = \sqrt{Var(X)} $$

3. Continuous Random Variables

• Probability Density Function (p.d.f.)

  A non-negative integrable function $f(x)$ is called the p.d.f. of a continuous random variable $X$ defined on an interval $S=(a,b)$ if it satisfies:

  1. $f(x) \ge 0$, for all $x \in S$.
  2. The area under the curve $y=f(x)$ over $S$ is 1. That is, $\int_S f(x) dx = 1$.

  The probability that $X$ takes a value in an interval $A$ is:

  $$ P[X \in A] = \int_A f(x) dx $$

  Note: The probability that $X$ takes any specific exact value $x$ is 0. That is, $P[X=x] = 0$.

• Cumulative Distribution Function (c.d.f.)

  The c.d.f. of a continuous random variable $X$ up to point $x$ is defined as the area under the p.d.f. curve from the lower limit to $x$:

  $$ F(x) = \int_a^x f(t) dt \quad \text{for } a < x < b $$

  Area under p.d.f representing Cumulative Distribution $F(x)$
• Extended Properties

  If $f(x)$ is defined on $(a, b)$, we extend it to $\mathbb{R}$ by defining $f(x) = 0$ for $x \le a$ and $x \ge b$. Consequently:

  • $\int_{-\infty}^t f(x) dx = 0$ for $t \le a$
  • For $x \ge b$: $\int_{-\infty}^t f(x) dx = \int_{-\infty}^a f(x) dx + \int_a^b f(x) dx + \int_b^t f(x) dx = 0 + 1 + 0 = 1$

  Thus, $F(t) = 0$ for $t \le a$ and $F(t) = 1$ for $t \ge b$.