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Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector X. It is defined component by component, as E[X] i = E[X i]. Similarly, one may define the expected value of a random matrix X with components X ij by E[X] ij = E[X ij].
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
For two jointly distributed real-valued random variables and with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values: [3] [4]: 119
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 ...
The variance of a random variable is the expected value of the squared deviation from the mean of , = []: = [()]. This definition encompasses random variables that are generated by processes that are discrete , continuous , neither , or mixed.
The expected value (mean) (μ) of a beta distribution random variable X with two parameters α and β is a function of only the ratio β/α of these parameters: [1]
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.
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by [] =. In light of the examples given below , this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes.