# Random variable sets

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A random variable set is a vector of Random variables , where do not necessarely have the same distribution

#### Two random variables

Two events X and Y are independent if:

For continuous probabilities, the previous equation is expressed as:

where and are the marginal density functions, defined as

Let Y have a probability greater than zero. The probability of the event *X* assuming that *Y* occurred is (*X* and *Y* may not be independent):

If X and Y are continuous:

#### Several random variables

Let be a vector of random variables. The mean and the standard deviation of the term are:

The covariance of the random variable and is defined as:

It can be shown that:

The covariance measures the relationship between two random variable. If , is likely to be greater (resp. lesser) than its mean value if is greater (resp. lesser) than its mean value. If , is likely to be lesser (resp. greater) than its mean value if is greater (resp. lesser) than its mean value. Independent random variables are uncorrelated, whereas uncorrelated random variables are not necessary independent.

The covariance matrix is defined as:

The correlation is expressed as:

The value of the correlation is in the range *[-1, 1]*. It is a scaled measure of the relationship among random variables

The probability density function of -dimensional Gaussian distribution can be determined using the covariance matrix and a vector containing the mean of each random variable (the term being the mean value of ):