Search results
Results from the WOW.Com Content Network
Probability generating functions are particularly useful for dealing with functions of independent random variables. For example: If , =,,, is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
This does not look random, but it satisfies the definition of random variable. This is useful because it puts deterministic variables and random variables in the same formalism. The discrete uniform distribution, where all elements of a finite set are equally likely. This is the theoretical distribution model for a balanced coin, an unbiased ...
An R package poibin was provided along with the paper, [13] which is available for the computing of the cdf, pmf, quantile function, and random number generation of the Poisson binomial distribution. For computing the PMF, a DFT algorithm or a recursive algorithm can be specified to compute the exact PMF, and approximation methods using the ...
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 ...
are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of i.i.d. random variables = = is a compound Poisson distribution. In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0.
Then if the random variables X i indicate the number of times outcome number i is observed over the n trials, the vector X = (X 1, ..., X k) follows a multinomial distribution with parameters n and p, where p = (p 1, ..., p k). While the trials are independent, their outcomes X i are dependent because they must sum to n.
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.
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount; Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution; Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS ...