Chapter 3.2_ Probability Distribution

A probability distribution is a mathematical function that provides the chances of different possible outcomes occurring in an experiment or random process.
It indicates how likely each outcome is.

Probability distributions can be categorized into two main types: discrete and continuous.

Definition: Discrete probability distributions deal with distinct and separate outcomes.
Example: Rolling a fair 6-sided die.
- Possible outcomes: {1, 2, 3, 4, 5, 6}.
- Probability Mass Function (PMF): Used to represent discrete distributions.
- Total sum of all probabilities in a discrete distribution equals 1.
Coin Flip Example:
- Flipping a coin results in a discrete distribution with chances of:
  - Heads: 50%
  - Tails: 50%
- In 20 flips, expect around 10 heads and 10 tails, but actual results may vary due to randomness.
Die Roll Example:
- Each face from 1 to 6 on a die has an equal probability of 1/6 or approximately 16.67%.
- In rolling a die 20 times, on average expect about 3 occurrences of each number.

For a single roll of a fair 6-sided die:
- Probability of any one number (i.e., P(X=x) for x = 1 to 6) = 1/6.
- Sum of all probabilities:
  - (1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6) = 6/6 = 1.

Definition: Continuous probability distributions deal with outcomes that can take any value within a given range.
Examples:
- Variables such as mass or velocity of an object.
Probability Density Function (PDF):
- Used to represent continuous distributions, typically visualized with smooth curves.
- The area under the curve within a specific range represents the probability of the outcome falling within that range.

PDF of a market index tracking the stock performance of 500 largest companies listed on U.S. stock exchanges.