Gaussian vs Normal Distribution
First and foremost the normal distribution and the Gaussian distribution are used to refer the same distribution, which is perhaps the most encountered distribution in the statistical theory.
For a random variable x with Gaussian or Normal distribution, the probability distribution function is P(x)=[1/(σ√2π)] e^(-(x-µ)2/2σ2 ); where µ is the mean and σ is the standard deviation. The domain of the function is (-∞,+∞). When plotted, it gives the famous bell curve, as often referred in social sciences, or a Gaussian curve in physical sciences. Normal distributions are a subclass of elliptical distributions. It can also be considered as a limiting case of the binomial distribution, where the sample size is infinite.
Normal distribution has very unique characteristics. For a normal distribution, the mean, the mode, and the median are the same, which is µ. The skewness and the kurtosis are zero, and it is the only absolutely continuous distribution with all the cumulants beyond the first two (mean and variance) are zero. It gives the probability density function with maximum entropy for any values of the parameters µ and σ2. The normal distribution is based on the central limit theorem, and it can be verified using practical results following the assumptions.
The normal distribution can be standardized using a transformation z=(X-µ)/σ, which converts it to a distribution with µ=0 and σ=σ2=1. This transformation allows easy reference to the standardized value tables and makes it easier to solve problems regarding the probability density function and the cumulative distribution function.
Applications of normal distribution can be categorized into three classes. Exact normal distributions, approximate normal distributions, and modeled or assumed normal distributions. Exact normal distributions occur in nature. The velocity of the high temperature or ideal gas molecules and ground state of the quantum harmonic oscillators show normal distributions. Approximate normal distributions occur in many cases explained by the central limit theorem. Binomial probability distribution and Poisson distribution, which are discrete and continuous respectively, show a likeness to normal distribution at very high sample sizes.
In practice, in a majority of the statistical experiments, we assume the distribution to be normal, and the model theory that follows is based on that assumption. As a result, the parameters can be readily calculated for the population and the inference process becomes easier.
What is the difference between Gaussian Distribution and Normal Distribution?
• Gaussian distribution and the Normal distribution are one and the same.