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Then, "independent and identically distributed" implies that an element in the sequence is independent of the random variables that came before it. In this way, an i.i.d. sequence is different from a Markov sequence , where the probability distribution for the n th random variable is a function of the previous random variable in the sequence ...
Not every sequence of random variables which converges to another random variable in distribution also converges in probability to that random variable. As an example, consider a sequence of standard normal random variables X n {\displaystyle X_{n}} and a second sequence Y n = ( − 1 ) n X n {\displaystyle Y_{n}=(-1)^{n}X_{n}} .
The extended versions of the theorem show that in any infinite sequence of exchangeable random variables, the random variables are conditionally independent and identically-distributed, given the underlying distributional form. This theorem is stated briefly below.
Consider a sequence (X n) n∈ of i.i.d. (Independent and identically distributed random variables) random variables, taking each of the two values 0 and 1 with probability 1 / 2 (actually, only X 1 is needed in the following). Define N = 1 – X 1.
A random variable X has a Bernoulli distribution if Pr(X = 1) = p and Pr(X = 0) = 1 − p for some p ∈ (0, 1).. De Finetti's theorem states that the probability distribution of any infinite exchangeable sequence of Bernoulli random variables is a "mixture" of the probability distributions of independent and identically distributed sequences of Bernoulli random variables.
[1] [2] [3] Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. [4]
One of the simplest stochastic processes is the Bernoulli process, [80] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability and zero with probability .
Let X 1, X 2, ... be a sequence of independent and identically distributed random variables, each with expected value 0 and finite variance, and let = + +. Then there is a sequence of stopping times τ 1 ≤ τ 2 ≤ ... such that the have the same joint distributions as the partial sums S n and τ 1, τ 2 − τ 1, τ 3 − τ 2, ... are independent and identically distributed random ...