What Is a Probability Distribution?
When working with statistics, you’ll often come across the term “probability distribution.” This concept highlights the overlap between probability and statistics. Despite sounding technical, a probability distribution is simply a way to organize a list of probabilities.
A probability distribution is a function or rule that assigns probabilities to each value of a random variable. It can be represented in a list or as a graph.
Definition
A probability distribution is a statistical function that describes all the possible values and likelihoods a random variable can take within a specified range. This range spans from the minimum to the maximum statistically possible values. The position of a value on the probability distribution depends on factors such as the distribution’s mean, standard deviation, skewness, and kurtosis.
For instance, if thirty-eight students take a test, the X-axis of a graph may show intervals of scores (e.g., the interval labeled 35 includes scores from 32.5 to 37.5), while the Y-axis shows the number of students scoring within or below each interval.
Cumulative Frequency Distribution
A cumulative frequency distribution can either show the actual frequencies at or below each interval or the percentage of scores at or below each interval. This can be displayed as a histogram or a polygon.
Types of Probability Distributions
Probability distributions are mathematical approaches to quantifying uncertainty. There are two main classes:
- Discrete Distributions: Describe variables that take on discrete values only (typically positive integers).
- Continuous Distributions: Describe variables that can take on any value within a continuum (typically real numbers).
Probability Distributions in Practice
It’s important to note that all the material covered in the course section using can also be implemented using MATLAB. These commands are frequently used in real-world applications, particularly in finance and engineering.
Ordered Subsets
Suppose we have a set of \(n\) items. From that set, we create a subset of \(k\) items. The order in which items are selected is recorded, making the ordering of selected items very important. The total number of ordered subsets of \(k\) items chosen from a set of \(n\) items is given by:
\[ \frac{n!}{(n-k)!} \]
Quantiles for Probability Distributions
The term quantile, first used by Kendall in 1940, refers to a number \(x_p\) such that a proportion \(p\) of the population values are less than or equal to \(x_p\).
- The 0.25 quantile (25th percentile or lower quartile) of a variable is a value \(x_p\) such that 25% of the values fall below that value.
- The 0.75 quantile (75th percentile or upper quartile) is a value \(x_p\) such that 75% of the values fall below that value.
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