### What is the formula for the probability density function of the normal distribution?

The formula for the probability density function of the normal distribution is f(x)=exp{-(x-μ)²/2σ²}/[√(2π)σ].

The normal curve is bell-shaped, low at both ends, high in the middle, and symmetrical from side to side because of its bell-shaped curve, so it is often called the bell curve. If the random variable x obeys a normal distribution with mathematical expectation and variance 0~2, denoted as N (µ, 02).

Its probability density function is the expectation of the normal distribution u determines its location, and its standard deviation mouth determines the magnitude of the distribution. The normal distribution when taking = 0, = 1 is the standard normal distribution.

Normal distribution curve

Normal distribution as the distribution of a continuous random variable with two parameters μ and σ^2, the first parameter μ is the mean of the random variable to comply with the normal distribution, and the second parameter σ2 is the variance of this random variable, so that the normal distribution is recorded as N (μ, σ^2).

The probability law of a random variable that follows a normal distribution is that the probability of taking a value close to μ is large, while the probability of taking a value farther away from μ is smaller; the smaller σ is, the more the distribution is concentrated near μ, and the larger σ is, the more the distribution is spread out. The density function of the normal distribution is characterized by symmetry about μ, a maximum value at μ, a value of 0 at positive (negative) infinity, and an inflection point at μ±σ.

Reference: Baidu Encyclopedia – Normal Distribution

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

The normal distribution is a probability distribution that is generally denoted by the symbols µ and σ for the mean and standard deviation. Its probability density function is:

f(x)=(1/σ√2π)exp(-(x-μ)²/2σ²)

Where μ is the mean of the normal distribution, σ is the standard deviation of the normal distribution, and e is a natural constant.

The standardized formula for the normal distribution is:

Y=(X-μ)/σ~N(0,1)

Where X is the original data, Y is the standardized data, μ is the mean of the original data, and σ is the standard deviation of the original data.

The Cumulative Distribution Function (CDF) of the normal distribution can be expressed as:

F(x)=(1+erf((x-μ)/√(2σ²))/2

Where, erf is the errorfunction, which can be obtained by looking up the table or the math library function.

### 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.

### How to calculate the density function of a normal distribution

The formula for the normal distribution function is P(x)=(2π)^(-1/2)*σ^(-1)*exp{[-(x-μ)^2]/(2σ^2)}. Where F(y) is the distribution function of Y and F(x) is the distribution function of X. where μ is the mean and σ is the standard deviation. μ determines the location of the normal distribution, and the closer it is to μ, the greater the probability that it will be taken, and vice versa.

σ describes the degree of dispersion of the normal distribution, the larger σ is, the more dispersed the data distribution is the flatter the curve is. the smaller σ is, the more concentrated the data distribution is the steeper the curve is. If the random variable X obeys a probability distribution with a location parameter of μ and a scale parameter of σσ, and its probability density function is f(x) = 12π – √σe – (x – μ)22σ2.

Characteristics of the normal distribution function

1. Concentration, the peak of the normal curve is located right in the center, where the mean is located.

2, symmetry, the normal curve is centered on the mean, symmetrical from side to side, and the ends of the curve never intersect the horizontal axis.

3, uniform change answerability, the normal curve from the mean is located in the beginning, respectively, to the left and right sides of the gradual and uniform decline.

4, the normal distribution has two parameters, namely, the mean μ and the standard deviation σ, can be written as N (μ, σ).

5, u transformation, in order to facilitate the description and application, often the normal variable for data transformation.