The normal curve and its properties
1. The normal distribution is often denoted as N(), and its normal distribution function: f(x)=, x∈(-∞,+∞).
N(0,1) is called the standard Normal distribution, corresponding function expression: f(x)=,x∈(-∞,+∞).
2. Properties of normal image:
①The curve is above the x-axis and does not intersect with the x-axis.
② The curve is symmetrical about the straight line x=μ.
③The curve is at the highest point when x=μ.
④When xμ, the curve drops, and when the curve extends infinitely to the left and right sides, the x-axis is the asymptote, and it goes infinitely to it ⑤When
μ is constant, the shape of the curve is determined. The larger the value, the more “squatty” the curve is, indicating that the overall distribution is more dispersed; the smaller the value, the more “thin and taller” the curve is, indicating that the overall distribution is more concentrated
. The transformation of general normal distribution and standard normal distribution
For the standard normal distribution, it is used to express the probability that the overall value is less than x0, that is, =p(x
Normal distribution, also known as "normal distribution", also known as
Gaussian
distribution , was first obtained by A. De Moffer
in
the asymptotic formula for binomial distribution. CF Gauss derived it from another angle when studying measurement error. PS Laplace and Gauss studied its properties. It is a probability distribution that is very important in
mathematics
,
physics and engineering, and has a significant impact on many aspects of statistics.
The normal curve is bell-shaped, low at both ends, high in the middle, and symmetrical to the left and right. Because the curve is bell-shaped, people often call it a
bell
-shaped curve .
If the
random
variable X obeys a normal distribution with
mathematical
expectation μ and
variance
σ^2, denoted as N(μ,σ^2). Its
probability
density function determines its location for the
expected value
, and its standard deviation σ determines the magnitude of the distribution. The normal distribution when μ = 0 and σ = 1 is the standard normal distribution .
one-dimensional normal distribution
If
random variable
Obeys a positional argument as
, the scale parameter is
The probability distribution of , and its
probability
density function is
[2]
Then this
random
variable is called a normal random variable, and the distribution obeyed by a normal random variable is called a normal distribution, denoted as
,read
obey
,or
obbey normal distribution.
When a μ-dimensional random
vector
has a similar probability law, the random vector is said to follow a multi-dimensional normal distribution. The multivariate normal distribution has good properties. For example, the marginal distribution of the multivariate normal distribution is still a normal distribution,
and
the random vector obtained by any linear transformation is still a multidimensional normal distribution, especially its linear combination is a univariate normal distribution. distributed.
The normal distribution of this entry is a one-dimensional normal distribution, and for multi-dimensional normal distribution, please refer to "
two-dimensional normal distribution
".
Standard normal distribution
when
When the normal distribution becomes the standard normal distribution
