what are the cdf and pdf the normal distribution

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certainly! let's delve into the concepts of the probability density function (pdf) and cumulative distribution function (cdf) of the normal distribution, along with code examples to illustrate these concepts.

understanding normal distribution

the normal distribution, often referred to as the gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. it is defined by two parameters: the mean (μ) and the standard deviation (σ).

**mean (μ)**: the central peak of the distribution.
**standard deviation (σ)**: measures the spread or dispersion of the distribution.

probability density function (pdf)

the pdf of a normal distribution gives the relative likelihood of a random variable to take on a given value. for a normal distribution, the pdf is given by the formula:

\[
f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}
\]

where:
\( f(x) \) is the value of the pdf at point \( x \).
\( e \) is the base of the natural logarithm.
\( \pi \) is a mathematical constant.

cumulative distribution function (cdf)

the cdf of a normal distribution represents the probability that a random variable \( x \) is less than or equal to a certain value \( x \). the cdf is given by the integral of the pdf from negative infinity to \( x \):

\[
f(x) = p(x \leq x) = \int_{-\infty}^{x} f(t) dt
\]

for the normal distribution, there isn't a closed-form expression for the cdf, but it can be computed using numerical methods or statistical libraries.

code example

let's use python with the `numpy` and `scipy` libraries to compute and plot the pdf and cdf of a normal distribution.



explanation of the code

1. **imports**: we import necessary libraries, including `numpy` for numerical operations, `matplotlib` for plotting, and `scipy.stats` for statistical functions.
2. **parameters**: we define the mean (`mu`) and standard deviation (`sigma`) of the normal distribution.
3. **generate x values**: we create an ...

#NormalDistribution #CDF #numpy
normal distribution
CDF
PDF
cumulative distribution function
probability density function
Gaussian distribution
statistical analysis
bell curve
standard normal distribution
probability theory
random variables
continuous distribution
mean
variance
distribution properties