Write down the formula and properties of a Normal Distribution.
Answers
Step-by-step explanation:
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
Normal Distribution
Probability density function

The red curve is the standard normal distribution
Cumulative distribution function
Notation{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}Parameters{\displaystyle \mu \in \mathbb {R} } = mean (location)
{\displaystyle \sigma ^{2}>0} = variance (squared scale)Support{\displaystyle x\in \mathbb {R} }PDF{\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}CDF{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}Quantile{\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2F-1)}Mean{\displaystyle \mu }Median{\displaystyle \mu }Mode{\displaystyle \mu }Variance{\displaystyle \sigma ^{2}}Skewness{\displaystyle 0}Ex. kurtosis{\displaystyle 0}Entropy{\displaystyle {\frac {1}{2}}\log(2\pi e\sigma ^{2})}MGF{\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}CF{\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}Fisher information{\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}} {\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}Kullback-Leibler divergence{\displaystyle D_{\text{KL}}({\mathcal {N}}_{0}\|{\mathcal {N}}_{1})={1 \over 2}\{(\sigma _{0}/\sigma _{1})^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+2\ln {\sigma _{1} \over \sigma _{0}}\}}
The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).
The probability density of the normal distribution is
{\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
where
{\displaystyle \mu } is the mean or expectation of the distribution (and also its median and mode),
{\displaystyle \sigma } is the standard deviation, and
{\displaystyle \sigma ^{2}} is the variance.
Step-by-step explanation:
The normal distribution is defined by the following equation: The Normal Equation. The value of the random variable Y is: Y = { 1/[ σ * sqrt(2π) ] } * e-(x - μ)2/2σ2. where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.14159, and e is approximately 2.71828.
Normal distributions are symmetric, unimodal, and asymptotic, and the mean, median, and mode are all equal.
A normal distribution is perfectly symmetrical around its center. That is, the right side of the center is a mirror image of the left side. There is also only one mode, or peak, in a normal distribution. Normal distributions are continuous and have tails that are asymptotic, which means that they approach but never touch the x-axis. The center of a normal distribution is located at its peak, and 50% of the data lies above the mean, while 50% lies below. It follows that the mean, median, and mode are all equal in a normal distribution.