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It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x,y), where z is a dependent variable and x and y are independent variables. [8] Functions with multiple outputs are often referred to as vector-valued functions.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
However, X, Y, and Z are not mutually independent, since ,, (,,) () (), the left side equalling for example 1/4 for (x, y, z) = (0, 0, 0) while the right side equals 1/8 for (x, y, z) = (0, 0, 0). In fact, any of {,,} is completely determined by the other two (any of X, Y, Z is the sum (modulo 2) of the others). That is as far from independence ...
Z is said to be a spurious variable and must be controlled for. The same is true for intervening variables (a variable in between the supposed cause (X) and the effect (Y)), and anteceding variables (a variable prior to the supposed cause (X) that is the true cause). When a third variable is involved and has not been controlled for, the ...
A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is =
A variable of this type is called a dummy variable. If the dependent variable is a dummy variable, then logistic regression or probit regression is commonly employed. In the case of regression analysis, a dummy variable can be used to represent subgroups of the sample in a study (e.g. the value 0 corresponding to a constituent of the control ...
In that model, the random variables X 1, ..., X n are not independent, but they are conditionally independent given the value of p. In particular, if a large number of the X s are observed to be equal to 1, that would imply a high conditional probability , given that observation, that p is near 1, and thus a high conditional probability , given ...
The degree of dependence between variables X and Y does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between X and Y, most correlation measures are unaffected by transforming X to a + bX and Y to c + dY, where a, b, c, and d are constants (b and d being positive).