7 Probability Distributions
A probability distribution describes how probabilities are assigned to the possible outcomes of a random variable. In other words, it shows how likely each value or range of values is to occur in an experiment or real-world process. Understanding probability distributions is critical for modeling uncertainty, making predictions, and performing statistical analyses.
There are two main types of probability distributions: discrete and continuous.
- Discrete probability distributions apply to variables that can take a finite or countable number of values. Examples include the number of heads when flipping a coin multiple times or the number of customers arriving at a store in an hour. Common discrete distributions include the Binomial and Poisson distributions.
- Continuous probability distributions apply to variables that can take any value within a range. Examples include height, weight, or time between arrivals. Continuous distributions are described using probability density functions (PDFs) rather than simple probabilities. The Normal (Gaussian) distribution is the most widely known continuous distribution.
Key properties of probability distributions include:
- Mean (Expected value): The long-run average value of the random variable.
- Variance and Standard Deviation: Measures of how much the values deviate from the mean.
- Shape of the distribution: Symmetry, skewness, and kurtosis describe the form of the distribution, influencing which statistical methods are appropriate.
Probability distributions (see Figure 7.1) link closely to earlier topics, including Central Tendency and Statistical Dispersion. For instance, the mean of a probability distribution corresponds to a measure of central tendency, while the variance or standard deviation quantifies its dispersion. Graphical tools such as histograms, density plots, and cumulative distribution plots help visualize these characteristics, making it easier to understand the behavior of random variables khan_academy_prob_dist?, statisticsbyjim_dist?, geeksforgeeks_prob_dist?.