In the realm of statistics, a probability distribution describes the probability of various potential results that can come from an experiment. It provides an understanding of the distribution of events in specific circumstances. Probability distributions can be distinguished between continuous and discrete based on whether they deal with numerical or categorical data. They are often involved in inferential statistics, risk assessment, and hypothesis testing.

## Definition: Probability distribution

Probability distributions are mathematical functions characterizing the likelihood that certain variable values will be attained. Graphs and probability tables are common ways to visualize probability distributions.

Binomial, Poisson, and uniform distributions are all examples of popular types of probability distributions. The ordinary, normal distribution, the **F distribution**, and the student’s **t distribution** are all employed in hypothesis testing. Probabilities of distributions are used to characterize the populations of real-world variables, such as the outcomes of coin flips and the masses of chicken eggs. P-values are also calculated from them when conducting hypothesis tests.

## Discrete probability distribution

Discrete likelihood distributions are variables that can only take on one of many possible values. Distributions of discrete probabilities only contain those values for which a possibility can be calculated.

### Probability tables

Probability tables are used to graphically depict the discrete distribution of categorical data. Two columns (class limits) and (their probabilities) make up the table.

### Probability mass function (PMF)

Discrete probability distributions may be modeled by means of a mathematical function known as a probability mass function (PMF). Essentially, it calculates the likelihood of each potential value for a variable. Consider the possibility that the average UK shirt collection follows a Poisson distribution. The formula gives the PMF:

is the likelihood that one has exactly k shirts. | |

is the average amount of sweaters owned by a population. | |

is Euler’s constant |

### Types of discreet probability distributions

Distribution |
Description |
Example |

Binominal |
Identifies factors with two distinct results. | The probability that, after five flips, a coin will come up heads every time. |

Discrete uniform |
Defines situations when the odds of happening are the same for everyone. | The typical distribution of playing card suits. |

Poisson |
It calculates the likelihood of k events occurring in a given period or place. | The daily number of texts. |

## Continuous probability distributions

A continuous variable admits an endless number of possible values. When a variable is considered continuous across a non-empty subset of the real numbers, it may take on any value inside that subset.

### Probability density functions (PDF)

PDFs are mathematical functions used to characterize continuous probability distributions. As such, it gives us the probability density for all possible values of a variable, even those that exceed one. It is possible to depict a probability density function as an equation or a graph.

A curve represents the probability density function in a graph. Determining the area under the curve within a certain range might help you estimate the likelihood that a given value will lie within that gap. Another sort of function used to characterize continuous probability distributions is the cumulative distribution function.

### Types of continuous probability distributions

With continuous data, many alternative distribution functions are possible. This set of handouts features:

Distribution |
Explanation |
Example |

Exponential |
Data is described in which a small value's likelihood is more significant than a great value. Time interval distribution is the likelihood of occurrence between unrelated events. | Intervals between earthquakes |

Continuous uniform |
Describes information when intervals of similar size have the same probability. | Time spent at a green light by vehicles. |

Normal distribution |
Provides a bell-shaped density function of the likelihood for data, where outlying numbers become less likely as the distance from the mean increases. | Analyzing SAT results |

Log-normal |
Simulates right-skewed patterns, often occurring when growth rates are assumed constant regardless of initial sample size. | Percentage information about body fat |

## Probability distribution: Expected value & standard deviation

Obtaining an equation, data or probability chart for a distribution allows one to calculate its estimated return and standard deviation. A distribution’s mean is also known as its anticipated value. As an expression, it is often represented as .

#### Expected value

Hyalines usually have a nest size of two to four eggs. Consider the following hypothetical probability distribution of Hyaline egg counts per nest.

Eggs (x) |
Probability (P(x)) |
x × P(x) |

2 | .2 | 2 × 0.2= 0.4 |

3 | .5 | 3 × 0.5= 1.5 |

4 | .3 | 4 × 0.3= 1.2 |

The expected value of eggs per net is the sum of all values after multiplying the number of eggs and probability.

#### Standard deviation

To find the standard deviation, compute every value’s deviation from the expected value. Then multiply the squared values by their probability:

Eggs (x) |
Probability (P(x)) |
x - E(x) |
[x – E(x)] 2 × P(x) |

2 | .2 | 2-3.1= -1.1 | (-1.1) 2 × 0.2 = 0.242 |

3 | .5 | 3-3.1= -0.1 | (-0.1) 2 × 0.5 = 0.005 |

4 | .3 | 4-3.1= 0.9 | (0.9) 2 × 0.3 = 0.243 |

Obtain the square root after adding the values.

## Probability distribution: Testing hypotheses with null distributions

Evaluating hypotheses often makes use of null distributions. When the test’s null hypothesis is accepted as true, a statistic’s likelihood falls inside a range called its null distribution.^{4} The sample is summed up in a single number called a test statistic, which is then compared to the null distribution to get a probability value.

If the null hypothesis is true, the p-value represents the likelihood of receiving a number equal to or more drastic than the sample’s statistical test.

### Testing hypotheses using null distributions

One-sample t-tests are hypothesis tests that use the usage of the test statistic known as Student’s t. In the null distribution of Student’s t, the p-value (for a one-sided test) is the shading region to the right of t = 1.7.

The area is equal to.06 and may be determined using a calculator, statistical software, or reference tables. As this is a rather small sample, the significance level is.06.

### Common null distributions

Distribution |
Statistical test |

F- distribution |
ANOVA Equality of two variances |

Chi-square |
Chi-square goodness of fit test Test of a single variance |

Standard normal |
One sample location test |

Student's t distribution |
Linear regression One sample t-test. Two sample t-tests |

## Probability distribution: Formulas

In the following, the formulas of the probability distribution will be listed:

Distribution |
Formula |
Type of formula |

Normal | Probability density function | |

Poisson | Probability mass function | |

Discrete Uniform | Probability mass function | |

Binominal | Probability mass function |

## FAQs

Probability distributions are mathematical functions that characterize the likelihood of a set of alternative values for a variable.

A toss of a coin is the most basic illustration. There are two potential results when flipping a coin:

- Heads
- Tails

The data in a normal distribution are spread out evenly throughout the board. There seems to be a core zone where most values are concentrated, with a gradual decrease in value as one moves outside that area.

There are two main types of probability distributions:

- Continuous
- Discrete