How to find the coefficients of a probability density function

How to find the coefficients of the probability density function and the coefficients of the distribution function (online, etc.)

Probability density function from negative infinity to positive infinity of the integral is 1, you can determine the coefficients

Distribution function when the variable tends to the negative infinity of the limit is 0, the positive infinity is the limit of the coefficients is 1, you can determine the coefficients.

How to find 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.

Probability density:

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

How to find the probability density function?

The Gaussian probability density function formula is composed of a univariate normal distribution, and a multivariate normal distribution.

Univariate Gaussian distribution:

Univariate Gaussian distribution probability density function is defined as:


In the formula μμ is the expectation of the random variable xx, σ2σ2 is the variance of xx, and σσ is known as the standard deviation:

μ=E(x)=∫∞∞xp( x)dx,


It can be seen that this probability distribution function, which is fully determined by the expectation and variance. The samples of Gaussian distribution are mainly concentrated around the mean, and the degree of dispersion can be expressed by the standard deviation, the larger it is, the greater the degree of dispersion, and about 95% of the samples fall in the interval (μ2σ,μ+2σ).

Multivariate Gaussian distribution:

Probability density function of a multivariate Gaussian distribution. Definition of the probability density function of the multivariate Gaussian distribution:


where x=[ x1,x2,… ,xd]Tx=[x1,x2,… ,xd]T is a dd-dimensional column vector;

μ=[μ1,μ2,… ,μd]Tμ=[μ1,μ2,… ,μd]T is a column vector of dd-dimensional means;

ΣΣ is a d × dd × d-dimensional covariance matrix;

Σ-1Σ-1 is the inverse matrix of ΣΣ;

|Σ||Σ| is the determinant of ΣΣ;

(x-μ) T(x-μ)T is the transpose of (x-μ)(x-μ) and


Σ=E{(x-μ)(x-μ)T}(2.3)(2.3)Σ =E{(x-μ)(x-μ)T}

Where μ,Σμ,Σ are the vectors xx and the matrices (x-μ)(x-μ)T(x-μ)(x -μ)T’s expectation, Noxixi is the iind component of xx, μiμi is the iind component of μμ, and σ2ijσij2 is the i,ji,jth element of ∑∑. Then: