The subject of random variables and probability distribution is a sub-area of stochastics in mathematics. It is dealt with in the upper school and is usually part of the Abitur material.
The Random Variables and Probability Distribution chapter is divided into four major thematic blocks, all of which you can find on our platform!
random variables
A random variable is a function X that assigns a real number to each outcome of a random experiment.
Formally:
First, in the chapter Random Variables and Probability Distribution, you will find the Random Variable section. There you will be explained what a random variable or random variable is and the difference between discrete and continuous random variables. You will also learn what the mean and variance of a random variable is and what you can think of as a histogram. The subject of the confidence interval or confidence level is also dealt with, as well as the independence of random variables.
probability distribution
A probability distribution or probability function of a random variable is a function associated with any value xi a random variable X a probability P(X=xi) assigned.
Formally:
First, this definition is explained in more detail in the article probability distribution. You will also learn about the density function and the distribution function, as well as the axioms of Kolmogorov, a Soviet mathematician who provided many ideas and proofs in probability theory. Finally, the term probability space is defined.
Discrete probability distribution
There are two different types of probability distributions. In this section you will learn what so-called discrete probability distributions are and get to know the most important of them.
Probability Distribution – Binomial Distribution
The binomial distribution according to Jakob Bernoulli describes the number of successes in a Bernoulli chain. A Bernoulli chain is a random experiment in which a Bernoulli experiment is performed n times in a row. A Bernoulli experiment is characterized by the fact that it only has the outcomes «success» and «failure».
Two common Bernoulli experiments are:
- flipping a coin with the result heads or tails.
- rolling a die with a result of «six» or «no six».
In a Bernoulli chain, the outcome of a single trial has no effect on subsequent trials. So every try is independently from all other attempts.
If a random variable X assumes the values k=0, 1, 2, …, n with the probabilities, then it is called binomially distributed.
This definition is deepened in the binomial distribution section. You can also read more about what a Bernoulli experiment and a Bernoulli chain are. You will also come across so-called three times at least problems and the spherical fan model, as well as get to know Laplace’s approximation formulas.
Poisson distribution
A random variable X is called Poisson distributedif for k=0, 1, 2, … the following applies: with .
The Poisson distribution is very important because under certain conditions it can be used to approximate the binomial distribution.
Hypergeometric Distribution
The hypergeometric distribution, like the binomial distribution, is used for random events where there are only two possible outcomes. The difference is that it accounts for varying probabilities of success and failure that may arise from attempt to attempt.
A random variable X is called hypergeometrically distributedif:
An urn contains 10 balls, 4 white and 6 black. The probability of drawing exactly 2 white balls when a total of 5 balls are drawn without replacement is given by:
The results of a lottery game are hypergeometrically distributed. Therefore, in this section you will also get to know the lottery formula, which you can use to calculate the probability of winning the lottery!
Continuous probability distribution
The second type of probability distribution is the continuous probability distribution. Here you will get to know three distributions:
Probability Distribution – Normal Distribution
The graph of the density function will also Gaussian bell curve called. If the expected value is 0 and the standard deviation is 1, this is called the standard normal distribution.
Figure 1: Standard normal distribution
The Gaussian function, named after the famous mathematician Gauss, and the sigma rules are also explained in the Normal Distribution section.
Probability distribution – uniform distribution
A random variable X is called uniform if every possible event occurs with the same probability. An example of this is rolling a dice. The numbers 1, 2, 3, 4, 5 and 6 all occur with probability.
Probability Distribution – Exponential Distribution
A random variable X such that , is exponentially distributed. where is a parameter.
The exponential distribution occurs, for example, in the service life of electrical devices. With such a device, for which the average service life is known, you can then calculate how high the probability is that it will break sooner or later.
Random Variables and Probability Distribution – The Most Important
- A random variable is a function X that assigns a real number to each outcome of a random experiment.
- A probability distribution or probability function of a random variable is a function that assigns a probability P(X=xi) to each value xi of a random variable X.
- If a random variable X assumes the values k=0, 1, 2, …, n with the probabilities, it is called binomial distributed.
- A random variable X is called Poisson distributedif for k=0, 1, 2, … the following applies:
- A random variable X is called hypergeometrically distributedif:
- If the probability density of a random variable X is given by the expected value and the standard deviation, it is called normally distributed.
- A random variable X for which is exponentially distributed.