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In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average.
The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by , then the mean is also known as the expected value of (denoted ()).
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical ...
A probability measure mapping the σ-algebra for events to the unit interval. The requirements for a set function μ {\displaystyle \mu } to be a probability measure on a σ-algebra are that: μ {\displaystyle \mu } must return results in the unit interval [ 0 , 1 ] , {\displaystyle [0,1],} returning 0 {\displaystyle 0} for the empty set and 1 ...
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values [15] (almost surely) [16] which means that the probability of any event can be expressed as a (finite or countably infinite) sum: = (=), where is a countable set with () =.
The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions. The mode is not necessarily unique in a given discrete distribution since the probability mass function may take the same maximum value at several points x 1, x 2, etc.