Define probability density function for normal variable.

In statistics and probability theory, the probability density function (PDF) of a continuous random variable describes the likelihood of the variable taking on a specific value within a given range. For a normal (Gaussian) random variable, the PDF is used to describe the distribution of the variable across the real number line.

The PDF of a normal random variable X with mean μ and standard deviation σ is given by the following formula:

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

where:

f(x) is the probability density function of the normal variable X at a given value x.
μ represents the mean (or expected value) of the normal distribution. It is the center of the distribution and indicates the average value of the variable.
σ is the standard deviation of the normal distribution. It measures the spread or dispersion of the variable's values around the mean.
π is the mathematical constant Pi (approximately 3.14159).
e is the base of the natural logarithm (approximately 2.71828).
Some key properties of the normal distribution:

The curve is symmetric around the mean μ.
The inflection points occur at μ - σ and μ + σ.
About 68.27% of the data falls within one standard deviation of the mean (i.e., between μ - σ and μ + σ).
About 95.45% of the data falls within two standard deviations of the mean (between μ - 2σ and μ + 2σ).
Approximately 99.73% of the data falls within three standard deviations of the mean (between μ - 3σ and μ + 3σ).
The normal distribution is widely used in various fields due to its mathematically tractable properties and its prevalence in nature and real-world data.