What is the probability density function a function of?
The function of density is the derivative. The density function f(x) is obtained by taking the derivative of x in the distribution function F(x). The density function f(x) is the derivative of the distribution function. In mathematics, the probability density function of a continuous random variable (which can be shortened to density function when not confusing) is a function that describes the likelihood that the output value of this random variable will be in the vicinity of some definite point of value.
Properties of the density function
The density function is non-negative and normalized. The probability density function of a continuous random variable (which can be shortened to density function when there is no confusion) is a function that describes the likelihood that the output value of this random variable will fall in the vicinity of some definite point of value. The probability that the value of the random variable falls within a certain region is the integral of the probability density function over this region.
What does probability density function mean?
The probability density function is for continuous random variables, suppose for a continuous random variable x, its distribution function is f(x) and its probability density is f(x).
First of all, for a continuous random variable x, its distribution function f(x) should be continuous, however, this function you gave is not continuous at the points x=-1, x=1, so there is no probability density function, maybe you made a mistake in solving the distribution function.
If f(x) is solved correctly, you can calculate the probability density along the following lines: by definition f(x)=∫[-∞,x].
f(y)dy can be seen f'(x)=f(x), that is, the derivative of the distribution function is equal to the probability density function, so you just need to find the derivative on the basis of the original distribution function to get the probability density function.
The probability distribution function is one of the basic concepts of probability theory. In practice, it is often necessary to study the probability that a random variable ξ takes a value less than a certain value of x, which is a function of x, called this function for the distribution function of the random variable ξ, referred to as the distribution function, notated as F (x), that is, F (x) = P (ξ & lt; x) (-∞ & lt; x & lt; + ∞), by which and can decide the probability of a random variable falling into any range.
For example, in the design of bridges and dams, the annual probability that the maximum water level ξ of the river is less than x meters is a function of x, this function is the distribution function of the maximum water level ξ. Commonly used distribution functions in practical applications are normal distribution function, Puasson distribution function, binomial distribution function and so on.
What is probability density function?
Density function refers to the probability density function.
The density function is the probability of a segment of an interval divided by the length of the interval, and the value is positive and can be large or small, while the distribution function is a curve that can be used to study random variables using mathematical analysis. The density function is generally only for continuous variables, while the distribution function is for both continuous and discrete random variables. The distribution function is solved using categorical discussions and definite integrals, and the density function is solved using derivatives.
The difference between the probability density and the distribution function is that the concepts are different, the objects described are different, and the solutions are different.
1, the concept of different: probability refers to the random occurrence of the event, for the uniform distribution function, the probability density is equal to the probability of a segment of the interval (the range of values of the event) divided by the length of the segment of the interval, which is non-negative, can be very large can also be very small; distribution function is an important function of probability statistics, it is through which, can be used in mathematical analysis of the method of study of random variables.
The distribution function is the most important probabilistic characteristic of a random variable, the distribution function can completely describe the statistical laws of a random variable and determine all other probabilistic characteristics of a random variable.
2, the description of different objects: the probability density is only for continuous variables, while the distribution function is a discussion of the probability of all random variables take values, including continuous and discrete.
3, the solution is different: the density function of a continuous random variable is known, the distribution function can be calculated through the discussion and the definite integral; when the distribution function of a continuous random variable is known, its derivatives can be obtained from the density function.
For discrete random variables, if you know the probability distribution (distribution series), you can also find its distribution function; of course, when you know the distribution function can also find the probability distribution.
What is the probability density function?
Set: the probability distribution function is: F(x)
The probability density function is: f(x)
The relationship between the two is: f(x)=dF(x)/dx
That is, the density function f is the first-order derivative of the distribution function F. Or the distribution function is the integral of the density function.
The distribution function is defined because in many cases we do not want to know the probability of something being at a particular value, but at best we want to know the probability of it being in a certain range, and so the concept of the distribution function is introduced.
And the probability density, if continuous at x. It is the distribution function F(x) that is derived from x. Conversely, knowing the probability density function, the distribution function can be derived by integrating negative infinity to x.
There is no practical significance in speaking simply of the probability density, which must be predicated on a definite bounded interval. You can think of the probability density as the vertical coordinate, the interval as the horizontal coordinate, the integral of the probability density to the interval is the area, and this area is the probability of the event occurring in this interval, the sum of all the areas is 1. So analyzing the probability density of a point alone does not have any significance, it has to have the interval as a reference and comparison.
Reference to the above: Baidu Encyclopedia – Probability Density
Keep it simple what is a probability density function?
The probability density function is a function that represents the probability of a continuous type of event. What we call the probability of an event in everyday life, events are categorized as discrete and continuous. For example, throwing colors, is a discrete type event, because there are only 6 possible outcomes, 1,2,3,4,5,6. This is discrete, each probability is 1/6. And the continuity of the event, for example, I take any one real number from the interval [0,1], there are infinite number of outcomes of the event, which is continuous, the probability of each outcome as a variable, and the outcome itself as an independent variable, such a function is the probability density function. Since it is the probability is always in the interval [0,1], the probability density function must be positive everywhere. The distribution function is the indefinite integral of the probability density function. Since the probability density function is positive everywhere, the distribution function is increasing.