Normal Distribution - MATLAB & Simulink (2024)

• Create a probability distribution object NormalDistribution by fitting a probability distribution to sample data (fitdist) or by specifying parameter values (makedist). Then, use object functions to evaluate the distribution, generate random numbers, and so on.

• Work with the normal distribution interactively by using the Distribution Fitter app. You can export an object from the app and use the object functions.

• Use distribution-specific functions (normcdf, normpdf, norminv, normlike, normstat, normfit, normrnd) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple normal distributions.

• Use generic distribution functions (cdf, icdf, pdf, random) with a specified distribution name ('Normal') and parameters.

$$y=f(x|\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-{(x-\mu )}^2}{2{\sigma }^2}},\text{for}x\in ℝ.$$

$$p=F(x|\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^x{e}^{\frac{-{(t-\mu )}^2}{2{\sigma }^2}}}dt,\text{for}x\in ℝ.$$

$$\Phi \left(x\right)=\frac{1}{2}\left(1-\text{erf}\left(-\frac{x}{\sqrt{2}}\right)\right)$$

$$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^{x}e{}^{-t^2}dt=}2\Phi \left(\sqrt{2}x\right)-1.$$

• Binomial Distribution — The binomial distribution models the total number of successes in n repeated trials with the probability of success p. As n increases, the binomial distribution can be approximated by a normal distribution with µ = np and σ² = np(1–p). See Compare Binomial and Normal Distribution pdfs.

• Birnbaum-Saunders Distribution — If x has a Birnbaum-Saunders distribution with parameters β and γ, then

  $$\frac{\left(\sqrt{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}-\sqrt{\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)}{\gamma }$$

  has a standard normal distribution.

• Chi-Square Distribution — The chi-square distribution is the distribution of the sum of squared, independent, standard normal random variables. If a set of n observations is normally distributed with variance σ², and s² is the sample variance, then (n–1)s²/σ² has a chi-square distribution with n–1 degrees of freedom. The normfit function uses this relationship to calculate confidence intervals for the estimate of the normal parameter σ² .

• Extreme Value Distribution — The extreme value distribution is appropriate for modeling the smallest or largest value from a distribution whose tails decay exponentially fast, such as, the normal distribution.

• Gamma Distribution — The gamma distribution has the shape parameter a and the scale parameter b. For a large a, the gamma distribution closely approximates the normal distribution with mean μ = ab and variance σ² = ab². The gamma distribution has density only for positive real numbers. See Compare Gamma and Normal Distribution pdfs.

• Half-Normal Distribution — The half-normal distribution is a special case of the folded normal and truncated normal distributions. If a random variable Z has a standard normal distribution, then $$X=\mu +\sigma \left|Z\right|$$ has a half-normal distribution with parameters μ and σ.

• Logistic Distribution — The logistic distribution is used for growth models and in logistic regression. It has longer tails and a higher kurtosis than the normal distribution.

• Lognormal Distribution — If X follows the lognormal distribution with parameters µ and σ, then log(X) follows the normal distribution with mean µ and standard deviation σ. See Relationship Between Normal and Lognormal Distributions.

• Multivariate Normal Distribution — The multivariate normal distribution is a generalization of the univariate normal to two or more variables. It is a distribution for random vectors of correlated variables, in which each element has a univariate normal distribution. In the simplest case, there is no correlation among variables, and elements of the vectors are independent, univariate normal random variables.

• Poisson Distribution — The Poisson distribution is a one-parameter discrete distribution that takes nonnegative integer values. The parameter, λ, is both the mean and the variance of the distribution. As λ increase, the Poisson distribution can be approximated by a normal distribution with µ = λ and σ² = λ.

• Rayleigh Distribution — The Rayleigh distribution is a special case of the Weibull distribution with applications in communications theory. If the component velocities of a particle in the x and y directions are two independent normal random variables with zero means and equal variances, then the distance the particle travels per unit time follows the Rayleigh distribution.

• Stable Distribution — The normal distribution is a special case of the stable distribution. The stable distribution with the first shape parameter α = 2 corresponds to the normal distribution.

  $$N\left(\mu ,{\sigma }^2\right)=S\left(2,0,\frac{\sigma }{\sqrt{2}},\mu \right).$$

• Student's t Distribution — The Student’s t distribution is a family of curves depending on a single parameter ν (the degrees of freedom). As the degrees of freedom ν goes to infinity, the t distribution approaches the standard normal distribution. See Compare Student's t and Normal Distribution pdfs.

  If x is a random sample of size n from a normal distribution with mean μ, then the statistic

  $$t=\frac{\overline{x}-\mu }{s/\sqrt{n}}$$

  where $$\overline{x}$$ is the sample mean and s is the sample standard deviation, has the Student's t distribution with n–1 degrees of freedom.

• t Location-Scale Distribution — The t location-scale distribution is useful for modeling data distributions with heavier tails (more prone to outliers) than the normal distribution. It approaches the normal distribution as the shape parameter ν approaches infinity.