Introduction to Probability Distributions

Probability Distributions

Definition: A probability distribution defines how the probabilities are distributed over the values of a random variable.
Types of Probability Distributions:
- Discrete Probability Distribution: Used for discrete random variables (e.g., countable outcomes).
  - Examples: Binomial distribution, Poisson distribution, Geometric distribution.
- Continuous Probability Distribution: Used for continuous random variables (e.g., outcomes that can take any value).
  - Examples: Normal distribution, Exponential distribution, Uniform distribution.
Key Characteristics:
- Probability Mass Function (PMF): Defines the probability of discrete random variables. Summation of PMF over all possible outcomes equals 1.
- Probability Density Function (PDF): Defines the probability of continuous random variables. The area under the PDF curve over its range equals 1.
Common Distributions:
- Binomial Distribution:
  - Models the number of successes in a fixed number of trials.
  - Defined by parameters (n) (number of trials) and (p) (probability of success).
- Normal Distribution:
  - Symmetrical, bell-shaped distribution defined by mean ((\mu)) and standard deviation ((\sigma)).
  - Empirical rule: Approximately 68% of data within one standard deviation, 95% within two, and 99.7% within three.
- Poisson Distribution:
  - Models the number of events occurring within a fixed interval of time/space given a constant mean rate and independence of events.
  - Characterized by parameter (\lambda) (average rate of occurrence).
Properties of Probability Distributions:
- Mean (Expected Value): The theoretical average of a random variable, calculated differently for discrete and continuous distributions.
- Variance: Measures the spread of the distribution, indicating how much values deviate from the mean.
- Standard Deviation: The square root of variance, providing a measure of dispersion with the same unit as the random variable.
Applications:
- Used in statistics, economics, engineering, and various fields for modeling random processes and making predictions based on data.
- Essential for hypothesis testing, confidence intervals, regression analysis, and quality control.
Important Concepts:
- Central Limit Theorem (CLT): States that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the shape of the original population distribution.
- Joint and Marginal Distributions: Joint distribution describes the probability of two variables occurring simultaneously; marginal distributions provide the probabilities of a subset of variables.
Calculation:
- For discrete distributions, use summation techniques.
- For continuous distributions, use integral calculus to compute probabilities and areas under the curve.

Probability Distributions

A probability distribution describes how probabilities are spread across different values of a random variable.
Discrete Probability Distributions are used for variables with countable outcomes.
- Examples include the Binomial, Poisson, and Geometric distributions.
Continuous Probability Distributions are used for variables with outcomes that can take any value within a range.
- Examples include the Normal, Exponential, and Uniform distributions.

Key Characteristics

Probability Mass Function (PMF) defines the probability of each individual outcome for discrete random variables.
- The sum of probabilities for all possible outcomes in a PMF is always 1.
Probability Density Function (PDF) defines the probability of a range of values for continuous random variables.
- The area under the PDF curve over its entire range is always 1.

Common Distributions

Binomial Distribution models the number of successes in a fixed number of independent trials.
- It is defined by two parameters: (n) (number of trials) and (p) (probability of success on each trial).
Normal Distribution is a symmetrical, bell-shaped distribution characterized by its mean ((\mu)) and standard deviation ((\sigma)).
- The Empirical Rule states that approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
Poisson Distribution models the number of events occurring within a fixed interval of time or space.
- It assumes a constant mean rate of occurrence and independence of events.
- It is defined by the parameter (\lambda) representing the average rate.

Properties of Probability Distributions

Mean (Expected Value) is the theoretical average of the random variable.
Variance measures the spread or variability of the distribution, indicating how much values deviate from the mean.
Standard Deviation is the square root of variance, providing a measure of dispersion in the same units as the random variable.

Applications

Probability distributions are widely applied in various fields like statistics, economics, engineering, and finance.
They are essential for:
- Modeling random processes.
- Making predictions based on data.
- Hypothesis testing.
- Confidence intervals.
- Regression analysis.
- Quality control.

Important Concepts

Central Limit Theorem (CLT) states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the original population distribution.
Joint and Marginal Distributions:
- Joint distribution describes the probability of two or more variables occurring simultaneously.
- Marginal distributions provide the probabilities of a subset of variables.

Calculation

Probabilities for discrete distributions are calculated using summation techniques.
Probabilities for continuous distributions are calculated using integral calculus to compute areas under the curve.