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.
Inhaltsverzeichnis
- 1 Probability Distribution – In a Nutshell
- 2 Definition: Probability distribution
- 3 Discrete probability distribution
- 4 Continuous probability distributions
- 5 Probability distribution: Expected value & standard deviation
- 6 Probability distribution: Testing hypotheses with null distributions
- 7 Probability distribution: Formulas
- 8 FAQs
Probability Distribution – In a Nutshell
- Two forms of data may be used in statistical analysis: discrete and continuous.
- A way to measure the possibility of an event is to look at its probability distribution.
- The parameters of a probability distribution specify its form.
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.
Example
A probability distribution table is relevant to use in order to characterize the range of outcomes from a single coin toss:
| Outcome | Probability |
| Heads | Tails |
| .5 | .5 |
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.
Example
Probability distribution idealized as a frequency distribution:
Consider a farmer interested in the probability of a specific sized egg being produced by her farm. The farmer takes 100 eggs at random and uses a histogram to explain the distribution of the eggs’ weights. The most reasonable approach is acknowledging that the size of eggs seems to follow a normal, standard distribution.
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.
Example
A robotic voice offers an unexpected welcome to humans. Below is a probability table that describes the distribution of greetings given certain conditions:
| Greetings | Probability |
| Hi! | .2 |
| Howdy! | .1 |
| Greetings, Human! | .7 |
Probability mass function (PMF)
Discrete probability distributions may be modelled 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.
Example
A regular distribution of egg weight may be described using the formula:
|
is the distribution of egg weight probabilities. |
|
is the average weight of a group's eggs (1.85 oz). |
|
defines the population's variation in egg weight (0.13 oz). |
It is statistically impossible for an egg to weigh precisely 2 ounces. A typical scale would say an egg weighs 2 ounces, but an infinitely exact scale might reveal minimal discrepancy.
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.
Example
eggs
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.
Example
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 |
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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
