Discrete probability distributions are mathematical models that describe the occurrence of events with discrete, finite values. These distributions are characterized by their properties, such as the sum of the probabilities of all possible outcomes equal to 1 and the presence of a parameter that determines the shape of the distribution. In this article, we will explore the characteristics of the most common discrete probability distributions, such as the Bernoulli distribution, the binomial distribution, the Poisson distribution, and the geometric distribution, as well as present some practical exercises to better understand these concepts.
Understanding the concept of discrete probability distribution: a simple and clear explanation.
To understand the concept of a discrete probability distribution, it's important to understand that it's a mathematical function that associates a probability with each possible outcome of a random experiment. In other words, the discrete probability distribution allows us to determine the chance of each outcome occurring in a finite or enumerable set of possibilities.
A discrete probability distribution is characterized by its probability function, which assigns each outcome a non-negative value, with the sum of all probabilities being equal to 1. Furthermore, the possible outcomes are distinct and isolated, with no possibility of intermediate values occurring.
A classic example of a discrete probability distribution is the Poisson distribution, widely used in counting processes, such as the number of events that occur in a given period of time. Another common example is the binomial distribution, which models experiments with only two possible outcomes, such as success or failure.
To apply the theory of discrete probability distributions, it is necessary to understand their specific properties and characteristics, as well as be able to calculate probabilities and interpret the results. Practical exercises are essential to deepen understanding and develop skills in this field of probability.
Learn about the main discrete distributions used in statistics and probability.
Learn about the main discrete distributions used in statistics and probability. Discrete probability distributions are important tools in statistical analysis, enabling the modeling and prediction of random events. Among the main discrete distributions are the Bernoulli distribution, the binomial distribution, the geometric distribution, the Poisson distribution, and the hypergeometric distribution.
A Bernoulli distribution is used to model experiments with only two possible outcomes, such as success and failure. binomial distribution It is applied in situations where there is a fixed number of independent trials, with only two possible outcomes in each trial, such as success and failure.
A geometric distribution is used to model the number of trials until the first success in a sequence of independent experiments. Poisson distribution is used to model the occurrence of rare events in a specific time or space interval.
Finally, the hypergeometric distribution It is used to model experiments in which there is a selection without replacement of elements from a finite population, with interest in the number of successes in a specific sample.
To better understand these discrete distributions and how to apply them, it's important to practice through exercises. Solving problems involving these distributions can help solidify knowledge and sharpen statistical and probability skills.
Therefore, when studying statistics and probability, it is essential to know the characteristics and applications of the main discrete distributions, such as the Bernoulli distribution, the binomial distribution, the geometric distribution, the Poisson distribution, and the hypergeometric distribution.
Types of probability distributions: learn about the different forms of statistical distributions.
Probability distributions are mathematical models that describe the random behavior of a phenomenon. There are different types of probability distributions, each with its own characteristics and applications. In this article, we will focus on discrete probability distributions, which are associated with discrete variables—those that can assume specific, countable values.
Some of the most common discrete probability distributions include the uniform distribution, the binomial distribution, the Poisson distribution, and the geometric distribution. Each of these distributions has its own properties and is used in different statistical contexts.
The uniform distribution, for example, is characterized by assigning the same probability to all possible values of a discrete variable. The binomial distribution is used to model the number of successes in a sequence of independent trials, each with the same probability of success. The Poisson distribution, in turn, is used to model the number of rare events in a time or space interval. And the geometric distribution is used to model the number of trials required until the first success in a sequence of independent trials.
To better understand how these distributions work, it's important to practice with exercises. For example, we can calculate the probability of getting exactly 3 heads in 5 tosses of a fair coin using the binomial distribution. Or we can determine the probability of at least 2 events occurring in a specific time interval using the Poisson distribution.
By understanding the characteristics and applications of these distributions, statistics and related science professionals can make more informed and accurate decisions based on probabilistic data.
Which variables are considered discrete in probability?
In probability, discrete variables are those that can assume a finite or countable number of values. This means that discrete variables are those that can be counted, usually represented by integers. For example, the number of cars in a parking lot, the number of students in a classroom, and the number of faces on a die are all examples of discrete variables.
These variables are distinct from continuous variables, which can assume an infinite number of values within a specific range. While discrete variables have specific, discrete values, continuous variables can assume any value within a continuous range. For example, a person's height, the time it takes to complete a task, and the room temperature are examples of continuous variables.
Therefore, discrete variables in probability are those that can be counted and take on specific, separate values, as opposed to continuous variables that can take on any value within a range.
Discrete Probability Distributions: Characteristics, Exercises
As discrete probability distributions are a function that associates to each element of X(S) = {x1, x2, …, xi, …}, where X is a given discrete random variable and S is the sampling space, the probability that this event happens. This function f of X(S) defined as f(xi) = P(X = xi) is sometimes called the mass probability function.
This probability mass is usually represented in the form of a table. Since X is a discrete random variable, X(S) has either a finite or an infinite number of events. Among the most common discrete probability distributions are the uniform distribution, the binomial distribution, and the Poisson distribution.
Features
The probability distribution function must meet the following conditions:
Furthermore, if X takes only a finite number of values (e.g., x1, x2, …, xn), then p(xi) = 0 if i > n and, therefore, the infinite series of conditions b becomes the finite series
This function also satisfies the following properties:
Let B be an event associated with the random variable X. This means that B is contained in X(S). Specifically, suppose that B = {xi1, xi2,…}. Therefore:
In other words: the probability of an event B is equal to the sum of the probabilities of the individual outcomes associated with B.
From this we can conclude that if the
PREMIUM QUALITY
Uniform distribution at n points
A random variable X is said to follow a distribution that is characterized by being uniform at n points if each value has the same probability assigned. Its probability mass function is:
Suppose we have an experiment with two possible outcomes: it could be flipping a coin whose possible outcomes are heads or tails, or choosing an integer whose outcome could be an odd or even number; This type of experiment is known as a Bernoulli test.
In general, the two possible outcomes are called success and failure, where p is the probability of success and 1-p is the probability of failure. We can determine the probability of x successes in n independent Bernoulli trials with the following distribution.
Binomial distribution
This function represents the probability of obtaining x successes in n independent Bernoulli trials, whose probability of success is p. Its probability mass function is:
The following graph represents the probability mass function for different values of the binomial distribution parameters.
The following distribution owes its name to the French mathematician Simeon Poisson (1781-1840), who obtained it as the limit of the binomial distribution.
Poisson Distribution
A random variable X is said to have a Poisson distribution of the parameter λ when it can receive the positive integer values 0,1,2,3, … with the following probability:
In this expression, λ is the average number of occurrences of the event for each unit of time and x is the number of times the event occurs.
Its mass probability function is:
Below is a graph representing the probability mass function for different values of the Poisson distribution parameters.
Note that as long as the number of successes is low and the number of tests performed on a binomial distribution is high, we can always approximate these distributions, since the Poisson distribution is the limit of the binomial distribution.
The main difference between these two distributions is that, while the binomial depends on two parameters – nep –, the Poisson depends only on λ, which is sometimes called the intensity of the distribution.
So far, we have only talked about probability distributions for cases where the different experiments are independent of each other; that is, when the outcome of one is not affected by the outcome of another.
When experiments are not independent, the hypergeometric distribution is very useful.
Hypergeometric distribution
Let N be the total number of objects in a finite set, of which we can identify k in some way, forming a subset K, whose complement is formed by the remaining Nk elements.
If we choose n objects at random, the random variable X representing the number of objects belonging to K in that choice will have a hypergeometric distribution of parameters N, n, and k. Its mass probability function is:
The following graph represents the probability mass function for different values of the hypergeometric distribution parameters.
Solved exercises
First exercise
Suppose the probability that a radio tube (placed in a certain type of equipment) will work for more than 500 hours is 0,2. If 20 tubes are tested, what is the probability that exactly k of them will work for more than 500 hours, k = 0, 1,2, 20, …, XNUMX?
Solution
If X is the number of tubes that run for more than 500 hours, we will assume that X has a binomial distribution. Then
And so:
For k≥11, the odds are less than 0,001
Thus, we can observe how the probability of k of these working more than 500 hours increases, until it reaches its maximum value (with k = 4) and then begins to decrease.
2nd exercise
A coin is flipped 6 times. When the result is heads, we call it a success. What is the probability of exactly two heads?
Solution
For this case, we have n = 6 and the probability of success and failure is p = q = 1/2
Therefore, the probability of two faces being given (i.e., k = 2) is
Third exercise
What is the probability of finding at least four faces?
Solution
For this case, we have k = 4, 5 or 6
Third exercise
Suppose 2% of the items produced in a factory are defective. Find the probability P that there are three defective items in a sample of 100 items.
Solution
For this case, we can apply the binomial distribution for n = 100 and p = 0,02, obtaining as a result:
However, since p is small, we use the Poisson approximation with λ = np = 2. Thus
References
- Kai Lai Chung: Elementary Probability Theory with Stochastic Processes. Springer-Verlag New York Inc.
- Kenneth.H. Rosen – Discrete Mathematics and its Applications. SAMCGRAW-HILL / INTERAMERICANO DE SPAIN.
- Paul L. Meyer Probability and statistical applications. SA ALHAMBRA MEXICANA.
- Seymour Lipschutz Ph.D. 2000 Solved Problems in Discrete Mathematics. McGraw-HILL
- Seymour Lipschutz Ph.D. Problems in Theory and Probability. McGraw-HILL