Discrete Distribution

Definition

A discrete distribution is a type of probability distribution that describes the likelihood of occurrence of each possible value of a discrete random variable. A discrete random variable is a variable that can only take on a countable number of distinct values, such as integers (0, 1, 2, 3...). Unlike continuous variables, there are "gaps" between the possible values, meaning the variable cannot take on every value within a range.

Main Content

1. Probability Mass Function (PMF)

• The PMF, denoted as $P(X = x)$, gives the probability that a discrete random variable $X$ is exactly equal to a specific value $x$.
• For a distribution to be valid, the sum of all probabilities for all possible values must equal 1 ($\sum P(x) = 1$) and each individual probability must be between 0 and 1.

2. Cumulative Distribution Function (CDF)

• The CDF, denoted as $F(x)$, calculates the probability that a random variable $X$ will take a value less than or equal to a specific value $x$.
• Mathematically, it is the running sum of the probabilities: $F(x) = P(X \leq x) = \sum_{t \leq x} P(t)$.

3. Expected Value and Variance

• The Expected Value $E(X)$ represents the long-term average outcome of the random variable, calculated as $\sum x \cdot P(x)$.
• The Variance $Var(X)$ measures the spread of the distribution around the mean, calculated as $E(X^2) - [E(X)]^2$.

Visual representation of a Discrete Probability Distribution:
Probability P(x)
  ^
0.4 |      [*]
0.3 |  [*]      [*]
0.2 |
0.1 |
    +-------------------> x (Value)
       0    1    2

The diagram above illustrates a sample space where outcomes 0, 1, and 2 have assigned probabilities.

Working / Process

1. Identify the Sample Space

• Determine all possible mutually exclusive outcomes for the experiment.
• Ensure that the list of outcomes is finite or countably infinite.

2. Assign Probabilities

• Determine the probability for each individual outcome based on theoretical logic or empirical observation.
• Verify that the sum of all these assigned probabilities equals 1.0.

3. Calculate Statistical Metrics

• Compute the mean (Expected Value) by multiplying each outcome by its probability and summing the results.
• Compute the Variance by finding the squared deviation from the mean for each outcome, multiplying by the outcome's probability, and summing these values.

Advantages / Applications

• Quality Control: Used in manufacturing to track the number of defective items in a batch (e.g., Binomial distribution).
• Insurance and Risk Management: Helps calculate the number of claims or accidents occurring in a specific time frame (e.g., Poisson distribution).
• Financial Modeling: Used to predict the frequency of specific financial events like stock market crashes or dividend payouts.

Summary

Discrete distribution is a probability model used for variables that take on distinct, separate values rather than a continuous range. It relies on the Probability Mass Function to assign likelihoods to specific outcomes, which must sum to one. These distributions are foundational tools for analyzing countable data, calculating expected averages, and managing risk in various academic and professional fields.

Important terms to remember: Random Variable, Probability Mass Function (PMF), Cumulative Distribution Function (CDF), Expected Value, Variance.