Normal distribution

Normal distribution - This article introduces the normal distribution. The normal distribution is of paramount importance in probability theory and statistics and it is widely employed to build models of real-world phenomena in both natural and social sciences. There are several reasons why the normal distribution is so important: 1. many random quantities that we observe in practice have a probability distribution that is approximately normal (for example, measurement errors in scientific experiments, men's height and weight, IQs); 2. the normal distribution naturally arises in the Central Limit Theorem (the sample mean converges to the normal distribution by increasing the sample size); 3. from a mathematical standpoint, the normal distribution is highly tractable (many properties of the normal distribution can be characterized through explicit mathematical formulae). The simplest case of a normal distribution is the standard normal distribution, i.e. the normal distribution with zero mean and unit variance. A random variable having a standard normal distribution is called a standard normal random variable. A standard normal random variable Z can be characterized by its probability density function f(z): f(z)=(2*pi)^(-1/2)*exp(-(1/2)*z^2) Since (2*pi)^(-1/2) is a constant, the probability density function f(z) is proportional to: exp(-(1/2)*z^2) Note that: 1. f(z) is maximized at z=0; therefore, the most likely outcome is Z=0;rn2. f(z) is symmetric; therefore values of opposite sign but equal magnitude have the same likelihood ( f(z)=f(-z) );rn3. the higher z is in absolute value, the lower f(z) is and the less likely it is that Z=z;rn4. the density has exponential decay; thus, it is very unlikely to observe realizations far out in the tails. As we have anticipated, a standard normal random variable has zero mean and unit variance. Any other normal random variable X that is not a standard normal random variable can be written as: X=mu+sigma*Z where Z is a standard normal random variable, mu is the mean of X and sigma^2 is its variance. Therefore, any normal random variable can be written as a linear transformation of a standard normal random variable. This means that any normal distribution is completely characterized by its mean mu and by its variance sigma. This is a very useful result: it means that all the theorems and propositions regarding the standard normal distribution (which are easier to derive) can be easily extended to non-standard normal distributions. You can find much more material about the normal distribution on StatLect, a free digital textbook on probability and statistics that provides access to an ever-growing collection of lectures and exercises on probability theory, statistics and econometrics. The main features of StatLect are the following: 1. step-by-step lectures: the essentials of each topic are first introduced in an intuitive manner and then, if necessary, they are repeated in a more rigorous manner; after that, more elementary details are added; finally, more advanced details are discussed. rn2. solved exercises and multiple choice tests: at the end of each lecture you can find solved exercises at various levels of difficulty (after trying to solve the problems, you can read a carefully explained solution); you can also find multiple choice tests scored in real time.rn2. hyperlinks: every time you find a technical term, you can jump to its definition by clicking on it;rn3. updated material: the lectures are continuously revised and updated, so that hopefully they become more complete and clearer and errors and typos are eliminated;rn4. ever-growing textbook: the number of lectures keeps growing. The digital textbook can be found at www.statlect.com.