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A Binomial distributed random variable X ~ B(n, p) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variables X ~ B(n, p) and Y ~ B(m, p) is equivalent to the sum of n + m Bernoulli distributed random variables, which means Z = X + Y ~ B(n + m, p). This can also be proven ...
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].
which is the mass function of a Poisson-distributed random variable with expected value λ. In other words, the alternatively parameterized negative binomial distribution converges to the Poisson distribution and r controls the deviation from the Poisson.
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance.
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, [1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability =.
The expected number of times the outcome i was observed over n trials is =. The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore
Consider the sum, Z, of two independent binomial random variables, X ~ B(m 0, p 0) and Y ~ B(m 1, p 1), where Z = X + Y.Then, the variance of Z is less than or equal to its variance under the assumption that p 0 = p 1 = ¯, that is, if Z had a binomial distribution with the success probability equal to the average of X and Y 's probabilities. [8]
The expected value of a Poisson random variable is ... The variance of the binomial distribution is 1 − p times that of the Poisson distribution, ...