various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution. Special distributions Normal distribution, the most common continuous distribution Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no) Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution, where the number of successes is one. Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die) Continuous uniform distribution, for continuously distributed values Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time Exponential distribution, for the time before the next Poisson-type event occurs Gamma distribution, for the time before the next k Poisson-type events occur Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test) Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test) Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution Statistical inference Statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation. Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations. Inferential statistics are used to test hypotheses and make estimations using sample data. Whereas descriptive statistics describe a sample, inferential statistics infer predictions about a larger population