Difference Between Probability Distribution Function and Probability Density Function

 

 

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Probability Distribution Function vs Probability Density Function
 

Probability is the likelihood of an event to happen. This idea is very common, and used frequently in the day to day life when we assess our opportunities, transaction, and many other things. Extending this simple concept to a larger set of events is a bit more challenging. For example, we cannot easily figure out the chances of winning a lottery, but it is convenient, rather intuitive, to say that there is a likelihood of one out of six that we are going get number six in a dice thrown.

When the number of events that can take place is becoming larger, or the number of individual possibilities is large, this rather simple idea of probability fails. Therefore, it has to be given a solid mathematical definition before approaching problems with higher complexity.

When the number of events that can take place in a single situation is large, it is impossible to consider each event individually as like in the example of the dice thrown. Hence, the whole set of events is summarized by introducing the concept of the random variable. It is a variable, which can assume the values of different events in that particular situation (or the sample space). It gives a mathematical sense to simple events in the situation, and mathematical way of addressing the event. More precisely, a random variable is a real value function over the elements of the sample space. The random variables can either be discrete or continuous. They are usually denoted by the uppercase letters of the English alphabet.

Probability distribution function (or simply, the probability distribution) is a function that assigns the probability values for each event; i.e. it provides a relation to the probabilities for the values that the random variable can take. The probability distribution function is defined for discrete random variables.

Probability density function is the equivalent of the probability distribution function for the continuous random variables, gives the likelihood of a certain random variable to assume a certain value.

If X is a discrete random variable, the function given as f(x) = P(X = x) for each x within the range of X is called the probability distribution function. A function can serve as the probability distribution function if and only if the function satisfies the following conditions.

1. f(x) ≥ 0

2. ∑ f(x) = 1

A function f(x) that is defined over the set of real numbers is called the probability density function of the continuous random variable X, if and only if,

P(a ≤ x ≤ b) = a∫b f(x) dx for any real constants a and b.

The probability density function should satisfy the following conditions too.

1. f(x) ≥ 0 for all x: -∞ < x < +∞

2. -∞∫+∞ f(x) dx = 1

Both probability distribution function and the probability density function are used to represent the distribution of probabilities over the sample space. Commonly, these are called probability distributions.

For statistical modeling, standard probability density functions and probability distribution functions are derived. The normal distribution and the standard normal distribution are examples of the continuous probability distributions. Binomial distribution and Poisson distribution are examples of discrete probability distributions.

What is the difference between Probability Distribution and Probability Density Function?

• Probability distribution function and probability density function are functions defined over the sample space, to assign the relevant probability value to each element.

• Probability distribution functions are defined for the discrete random variables while probability density functions are defined for the continuous random variables.

• Distribution of probability values (i.e. probability distributions) are best portrayed by the probability density function and the probability distribution function.

• The probability distribution function can be represented as values in a table, but that is not possible for the probability density function because the variable is continuous.

• When plotted, the probability distribution function gives a bar plot while the probability density function gives a curve.

• The height/length of the bars of the probability distribution function must add to 1 while the area under the curve of the probability density function must add to 1.

• In both cases, all the values of the function must be non-negative.

 

 

 

확률 분포 함수 대 확률 밀도 함수
 

확률은 사건이 일어날 가능성입니다. 이 아이디어는 매우 일반적이며 일상 생활에서 기회, 거래 및 기타 여러 가지를 평가할 때 자주 사용됩니다. 이 간단한 개념을 더 큰 이벤트 집합으로 확장하는 것은 조금 더 어렵습니다. 예를 들어, 우리는 복권에 당첨될 확률을 쉽게 파악할 수 없지만 던진 주사위에서 6번 중 1번이 6번이 될 가능성이 있다고 말하는 것이 편리하고 오히려 직관적입니다.

일어날 수 있는 사건의 수가 많아지거나 개별 가능성의 수가 많아지면 이 단순한 확률 개념은 실패합니다. 따라서 더 복잡한 문제에 접근하기 전에 확실한 수학적 정의가 주어져야 합니다.

하나의 상황에서 일어날 수 있는 사건의 수가 많으면 주사위를 던진 예와 같이 각 사건을 개별적으로 고려하는 것은 불가능하다. 따라서 전체 이벤트 집합은 확률 변수의 개념을 도입하여 요약됩니다. 특정 상황(또는 표본 공간)에서 다른 이벤트의 값을 가정할 수 있는 변수입니다. 상황의 단순한 사건에 수학적 의미를 부여하고 사건을 다루는 수학적 방법을 제공합니다. 더 정확하게 말하면, 확률 변수는 표본 공간의 요소에 대한 실제 값 함수입니다. 확률 변수는 이산적이거나 연속적일 수 있습니다. 그들은 일반적으로 영어 알파벳의 대문자로 표시됩니다.

확률 분포 함수(또는 간단히 확률 분포)는 각 이벤트에 대한 확률 값을 할당하는 함수입니다. 즉, 확률 변수가 취할 수 있는 값의 확률에 대한 관계를 제공합니다. 확률 분포 함수는 이산 확률 변수에 대해 정의됩니다.

확률 밀도 함수는 연속 확률 변수에 대한 확률 분포 함수와 동일하며, 특정 확률 변수가 특정 값을 가정할 가능성을 제공합니다.

X가 이산 확률 변수인 경우 X 범위 내의 각 x에 대해 f(x) = P(X = x)로 주어진 함수를 확률 분포 함수라고 합니다. 함수는 다음 조건을 충족하는 경우에만 확률 분포 함수로 사용할 수 있습니다.

1. f(x) ≥ 0

2. ∑ f(x) = 1

실수 집합에 대해 정의된 함수 f(x)는 다음과 같은 경우에만 연속 확률 변수 X의 확률 밀도 함수라고 합니다.

실수 상수 a와 b에 대한 P(a ≤ x ≤ b) = a∫b f(x) dx.

확률 밀도 함수는 다음 조건도 만족해야 합니다.

1. 모든 x에 대해 f(x) ≥ 0: -∞ < x < +∞

2. -∞∫+∞ f(x) dx = 1

확률 분포 함수와 확률 밀도 함수는 모두 표본 공간에 대한 확률 분포를 나타내는 데 사용됩니다. 일반적으로 이를 확률 분포라고 합니다.

통계 모델링을 위해 표준 확률 밀도 함수와 확률 분포 함수가 도출됩니다. 정규 분포와 표준 정규 분포는 연속 확률 분포의 예입니다. 이항 분포와 푸아송 분포는 이산 확률 분포의 예입니다.

확률 분포와 확률 밀도 함수의 차이점은 무엇입니까?

• 확률 분포 함수와 확률 밀도 함수는 각 요소에 해당 확률 값을 할당하기 위해 표본 공간에 대해 정의된 함수입니다.

• 확률 분포 함수는 이산 확률 변수에 대해 정의되고 확률 밀도 함수는 연속 확률 변수에 대해 정의됩니다.

• 확률 값의 분포(즉, 확률 분포)는 확률 밀도 함수와 확률 분포 함수에 의해 가장 잘 나타납니다.

• 확률분포함수는 표의 값으로 표현될 수 있지만, 확률밀도함수는 변수가 연속적이기 때문에 불가능하다.

• 플롯할 때 확률 분포 함수는 막대 플롯을 제공하고 확률 밀도 함수는 곡선을 제공합니다.

• 확률 분포 함수 막대의 높이/길이는 1을 더해야 하고 확률 밀도 함수의 곡선 아래 면적은 1을 더해야 합니다.

• 두 경우 모두 함수의 모든 값은 음수가 아니어야 합니다.

 

 

 

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