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This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
Given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter.
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /; French pronunciation:) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
The maximum likelihood estimator of is the value that maximizes the likelihood function given a sample. [16]: 308 By finding the zero of the derivative of the log-likelihood function when the distribution is defined over , the maximum likelihood estimator can be found to be ^ = ¯, where ¯ is the sample mean. [18]
The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions , i.e. the distributions of each of the individual random variables and the conditional probability distributions , which deal with how the outputs of one random variable are distributed when ...
Also, the characteristic function of the sample mean X of n independent observations has characteristic function φ X (t) = (e −|t|/n) n = e −|t|, using the result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself.
The "unexplained" component ( [|]) is simply the average of all the variances of within each group. The "explained" component Var ( E [ Y | X ] ) {\displaystyle \operatorname {Var} (\operatorname {E} [Y|X])} is the variance of the expected values, i.e., it represents the part of the variance that is explained by the variation of ...
In the special case that all possible outcomes are equiprobable (that is, p 1 = ⋅⋅⋅ = p k), the weighted average is given by the standard average. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.