Probability Distribution Overview

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1. Probability Distribution
   1. Random Variable
   2. Probability Distribution
   3. Binomial Distribution
   4. Poisson Distribution
   5. Normal Distribution </aside>

Probability Distribution

1. Random Variable

   1. A random variable is a numerical description of the outcome of an experiment. It allows us to map outcomes of random experiments to numbers for further analysis.
   2. Types of Random Variables:
      1. Discrete Random Variable:
         1. A random variable that can take on either a finite number of values or an infinite sequence of distinct values (e.g., 0, 1, 2, …).
         2. Example: The number of cars sold in a day (0, 1, 2, …).
      2. Continuous Random Variable:
         1. A random variable that can assume any numerical value within a given interval or collection of intervals.
         2. Example: The exact height of a randomly chosen person in a population (e.g., 165.4 cm, 170.2 cm).

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2. Probability Distribution of a Random Variable

   1. A probability distribution describes how probabilities are distributed across the values of the random variable. It provides a structured way to understand the likelihood of different outcomes.
   2. Key Terms:
      1. Probability Function (for Discrete Random Variables):
         1. A function, denoted as $f(x)$, that provides the probability for each value of the discrete random variable $x$.
         2. Example: When rolling a die, $f(x) = \frac{1}{6}$ for each possible outcome $x$ (1, 2, 3, 4, 5, 6).
      2. Probability Density Function (for Continuous Random Variables):
         1. A function, denoted as $f(x)$, that describes the likelihood of a continuous random variable falling within a particular interval.
         2. The probability that a continuous random variable equals a specific value is $0$, as the area under the curve at a single point is $0$.
         3. Example: The likelihood that a person’s height falls between 160 cm and 170 cm can be calculated by integrating the probability density function over this interval.

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3. Discrete Probability Function

   1. A discrete probability function must satisfy the following two conditions:
   2. $f(x) \geq 0$ for all possible values of $x$. (Probabilities cannot be negative.)
   3. $\sum f(x) = 1$, i.e., the sum of probabilities for all possible values of $x$ must equal 1. (The total probability of all outcomes is 100%.)
   4. Example:

      1. Consider the roll of a die.

      2. Possible outcomes: $x = \{1, 2, 3, 4, 5, 6\}$.

      3. Each outcome has an equal probability: $f(x) = \frac{1}{6}$.

      4. The probability distribution can be represented as:

         $x$	$f(x)$
         1	$\frac{1}{6}$
         2	$\frac{1}{6}$
         3	$\frac{1}{6}$
         4	$\frac{1}{6}$
         5	$\frac{1}{6}$
         6	$\frac{1}{6}$

      5. Verification of Conditions:

         1. $f(x) \geq 0$:

            $$ f(x) = \frac{1}{6} \ \text{is non-negative.} $$

         2. $\sum f(x) = 1$:

            $$ \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1 $$

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4. Continuous Probability Function

   1. A continuous probability function is defined over intervals, and the probabilities are computed as the area under the curve of the probability density function $f(x)$.

   2. Key Properties:

      $$ f(x) \geq 0 \text{ for all x } $$

   3. The total area under the curve of $f(x)$ over the interval equals 1:

      $$ \int_{-\infty}^\infty f(x) \, dx = 1 $$

   4. Example:

      1. The height of individuals in a population might follow a normal distribution (bell curve). If $X$ represents height, the probability that a randomly selected person has a height between 160 cm and 170 cm is computed as:

         $$ P(160 \leq X \leq 170) = \int_{160}^{170} f(x) \, dx $$

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5. Graphical Representation

   1. Example: Number of Cars Sold During a Day

      1. Suppose a car dealership tracks the number of cars sold in a day, with the following probability distribution:

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      2. Verification of Conditions:

      3. Graphical Representation:

      4. Summary of Required Conditions