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The sum of n geometric random variables with probability of success p is a negative binomial random variable with parameters n and p. The sum of n exponential (β) random variables is a gamma (n, β) random variable. Since n is an integer, the gamma distribution is also a Erlang distribution.
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3.
Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.
The exponential distribution is the continuous analogue of the geometric distribution. Applying the floor function to the exponential distribution with parameter λ {\displaystyle \lambda } creates a geometric distribution with parameter p = 1 − e − λ {\displaystyle p=1-e^{-\lambda }} defined over N 0 {\displaystyle \mathbb {N} _{0}} .
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
1.2 Geometric series, exponential function and sine. ... the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction.
The only continuous random variable that is memoryless is the exponential random variable. It models random processes like time between consecutive events. [8] The memorylessness property asserts that the amount of time since the previous event has no effect on the future time until the next event occurs.
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a ...