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# 数学代写|概率论代考Probability Theory代写|MS-E1600 RECORD TIMES AND RECORD VALUES IN THE CASE OF CONTINUOUS DISTRIBUTIONS

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## 数学代写|概率论代考Probability Theory代写|RECORD TIMES AND RECORD VALUES IN THE CASE OF CONTINUOUS DISTRIBUTIONS

Let us begin to consider the classic situation when $X_l, X_2, \ldots$ are independent random variables with a common continuous distribution function $\mathrm{F}(\mathrm{x})$. Let $\mathrm{L}(1)=1<\mathrm{L}(2)<\ldots<\mathrm{L}(\mathrm{n})<\ldots$ and $X(1)=X_I<X(2)<\ldots$ be the corresponding upper record times and upper record values. To work with such records it is convenient to attract the record indicators $\eta_n, n=1$, $2, \ldots$, which were mentioned above. For reasons of symmetry it is clear in this situation that with equal chances any of $\mathrm{n}$ random variables $\mathrm{X}1, \mathrm{X}_2, \ldots$, $\mathrm{X}{\mathrm{n}}$ can be maximal. It means that the following equality
$$\mathrm{P}\left{\eta_n=1\right}=1-\mathrm{P}\left{\eta_n=0\right}=1 / \mathrm{n}$$
is valid for any $\mathrm{n}=1,2, \ldots$
There is also the following relation, which is fair for continuous $\mathrm{X}$ ‘s:
$$\mathrm{P}\left{\eta_1=1, \eta_2=1, \ldots, \eta_n=1\right}=1 / \mathrm{n} ! .$$

## 数学代写|概率论代考Probability Theory代写|RECORDS IN THE SEQUENCES OF DISCRETE RANDOM VARIABLES

For simplicity of our exposition, below we will consider the initial identically distributed random variables $X_l, X_2, \ldots$, taking nonnegative integer values. The relations obtained for records in such sequences, relatively easily transferred to the case when identically distributed X’s have any arbitrary discrete distribution.

Let $X_l, X_2, \ldots$ take the values $n=0,1,2, \ldots$ with probabilities $p_n=P\left{X_k=\right.$ $n}, n=0,1,2, \ldots, k=1,2, \ldots$.

It should be noted that in such sequences of X’s it is possible with nonzero probabilities to get the situations, when the next observed value coincides with the last fixed record. Therefore, usually two discrete type of record values are considered. One of these schemas is associated with the study of strong records when the repetition of the previous record value is not recorded as a new record. However, in some situations the weak record values also present the subject of interest. It has been noted already that, for example, in a number of sport competitions the athlete, who repeated the existing record result, is announced as the corekordholder. Methods used in the study of the weak records, not very significantly different from the similar methods for the strong records. We therefore confine ourselves to the more detailed acquaintance with the strong record values $\mathrm{X}(1)=\mathrm{X}_1<$ $\mathrm{X}(2)<\ldots$

It should be noted immediately that the record indicators, when we study discrete records, also play a very important role. But it is necessary to mention that these indicators are different from the indicators used above. It should be reminded that for records in sequences of continuously distributed $\mathrm{X}$ ‘s the corresponding indicators $\eta_1, \eta_2, \ldots$ are defined in such a way that $\eta_n$ $=1$ if $\mathrm{n}$ is one of record times, and $\eta_n=0$ if $X_{\mathrm{n}}<\max \left{X_l, X_2, \ldots, X_{n-1}\right}$. For the considered discrete distributions we define another indicators $\mu_0, \mu_l, \ldots$. In this situation $\mu_n=1$, if the fixed number $\mathrm{n}$ represents some observed value in the sequence $X_l, X_2, \ldots$, which exceeds all values of the previous $\mathrm{X}$ ‘s, i.e., in this case there is such $\mathrm{m}=1,2, \ldots$, that $\mathrm{X}(\mathrm{m})=\mathrm{n}$. It appeared that in this scheme indicators $\mu_0, \mu_l, \ldots$ also have the important property that makes them comfortable in the study of discrete records. The following result is valid.

# 概率论代写

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## 数学代写|概率论代考Probability Theory代写|RECORDS IN THE SEQUENCES OF DISCRETE RANDOMVARIABLES

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