Convergence in Distribution
| Dice factory | |
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| Suppose a new dice factory has just been built. The first few dice come out quite biased, due to imperfections in the production process. The outcome from tossing any of them will follow a distribution markedly different from the desired uniform distribution. As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced dice will follow the uniform distribution more and more closely. |
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| Tossing coins | |
| Let Xn be the fraction of heads after tossing up an unbiased coin n times. Then X1 has the Bernoulli distribution with expected value μ = 0.5 and variance σ2 = 0.25. The subsequent random variables X2, X3, … will all be distributed binomially. As n grows larger, this distribution will gradually start to take shape more and more similar to the bell curve of the normal distribution. If we shift and rescale Xn’s appropriately, then will be converging in distribution to the standard normal, the result that follows from the celebrated central limit theorem. |
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| Graphic example | |
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Suppose { Xi } is an iid sequence of uniform U(−1,1) random variables. Let be their (normalized) sums. Then according to the central limit theorem, the distribution of Zn approaches the normal N(0, ⅓) distribution. This convergence is shown in the picture: as n grows larger, the shape of the pdf function gets closer and closer to the Gaussian curve. |
With this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution.
Convergence in distribution is the weakest form of convergence, since it is implied by all other types of convergence mentioned in this article. However convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.
Read more about this topic: Convergence Of Random Variables
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