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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.
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product = is a product distribution.
If more than one random variable is defined in a random experiment, it is important to distinguish between the joint probability distribution of X and Y and the probability distribution of each variable individually. The individual probability distribution of a random variable is referred to as its marginal probability distribution.
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 ...
In the event that the variables X and Y are jointly normally distributed random variables, then X + Y is still normally distributed (see Multivariate normal distribution) and the mean is the sum of the means. However, the variances are not additive due to the correlation. Indeed,
Two random variables and are conditionally independent given a random variable if they are independent given σ(W): the σ-algebra generated by . This is commonly written: This is commonly written: X ⊥ ⊥ Y ∣ W {\displaystyle X\perp \!\!\!\perp Y\mid W} or
Independent: Each outcome will not affect the other outcome (for from 1 to 10), which means the variables , …, are independent of each other. Identically distributed : Regardless of whether the coin is fair (with a probability of 1/2 for heads) or biased, as long as the same coin is used for each flip, the probability of getting heads remains ...