### How do you calculate the mean and variance of a normal distribution?

In a normal distribution, the mean is the center of the data and indicates the average value of the data; the variance is the degree of dispersion of the data and indicates the degree of dispersion of the data.

The formulas for calculating the mean and variance of a normal distribution are as follows:

Mean: μ = ∑x_i/n

Variance: σ^2 = ∑(x_i – μ)^2/(n-1)

Where x_i denotes the ith data in the sample, n denotes the number of data in the sample, μ denotes the mean, and σ^2 the variance.

For example, for a set of data {3,4,5,6,7}, the mean and variance are calculated as follows:

Mean: μ=(3+4+5+6+7)/5=5

Variance: σ^2=[(3-5)^2+(4-5)^2+(5-5)^2+(6-5)^2+(7-5)^2]/(5-1)=2< /p>

So, for this set of data, the mean is 5 and the variance is 2.

### What is the probability density function formula?

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:

p(x)=12πσ√exp{12(xμσ)2}

where μμ is the expectation of the random variable xx, σ2σ2 is the variance of xx, and σ σ is called the standard deviation:

μ=E(x)=∫∞∞xp( x)dx,

σ2=∫∞∞(xμ)2p(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:

p(x)=1(2π)d2|Σ|12exp{-12(x-μ)TΣ-1(x-μ)}

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

μ=E(x)

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:

μi=E(xi)=∫∞-∞xip(xi)dxi

### How do you find the probability density function of a normal distribution?

The probabilitydensityfunction (PDF) of a normal distribution (also known as a Gaussian distribution) is shown below:

f(x)=(1/(σ*√(2π)))*e^(-(x-μ)^2/(2σ^2))

In this formula:

-x is the value of the random variable;

-μ is the mean (expected value) of the normal distribution, which determines where the center of the distribution is located;

-σ is the standard deviation of the normal distribution, which determines the shape of the distribution, with the larger the standard deviation, the flatter the curve.

In the formula, e is the base of the natural logarithm (equal to about 2.71828) and π is the circumference.

The probability density function of a normal distribution describes the probability density of the variable taking on each value. The curve is bell-shaped, symmetrical about the mean, presents high points around the mean, and the probability density decreases as the distance from the mean increases.

It is important to note that the total area of the normal distribution is equal to 1, i.e., the probability densities under the entire curve sum to 1. This means that the probabilities for a particular range of values can be calculated by integrating the probability density function.