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Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Y i are independent observations from a normal distribution , Cochran's theorem shows that the unbiased sample variance S 2 follows a scaled chi-squared distribution (see also: asymptotic ...
The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable X, one defines the positive and negative parts by X + = max(X, 0) and X − = −min(X, 0). These are nonnegative random variables, and it can be directly checked that X = X + − X −.
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related.
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. [1] The term 'random variable' in its mathematical definition refers to neither randomness nor variability [ 2 ] but instead is a mathematical function in which
A distinction must be made between (1) the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and (2) the sample covariance, which in addition to serving as a descriptor of the sample, also serves as an estimated value of the population parameter.
A more general case of this concerns the distribution of the product of a random variable having a beta distribution with a random variable having a gamma distribution: for some cases where the parameters of the two component distributions are related in a certain way, the result is again a gamma distribution but with a changed shape parameter ...
Given two random variables X and Y, with expected values () = and () =, the expected value of random variable () (), written in statistical notation as (,). The covariance is used for measuring correlation ; it can be interpreted as the degree to which the two variables change simultaneously with each other or "co-vary".
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons , nats or hartleys) obtained about one random variable by observing the other random variable.