Order Statistics: CDF Calculation for i.i.d. Random Variables

In summary: The last integral should be ##f_X(x_3) \left[1-F_X(x_3)\right]^{K-3}##, but I don't know how to write it as a function of ##x_1##, ##x_2##, and ##x_3##.In summary, the conversation discusses the calculation of ##\text{Pr}\left[\sum_{i=1}^3Z_k\leq \eta\right]## where ##Z_k=X_k/Y_k## and ##X_k
  • #36
The problem was that ux was not reset to x_prev_lim for each uy value, and there was a mistake in the MC simulation. Now it works perfectly, and agrees with the MC simulation. Thanks for your responses @andrewkirk.

For the integration over Xs, dx is constant, and thus applying Simpson's rule for some optimization is straightforward. We simply compute dx = (uy*z - x_prev_lim)/L, where L is an even number (say 600) that represents the number of points in the numerical integration.

However, for Ys the step size is changing within the same integral. How can we deal with the Simpson's rule here, to make sure that the number of points is even?
 
<h2> What is the definition of order statistics?</h2><p>Order statistics refers to the collection of random variables that represent the ordered values of a sample from a population. In other words, it is the process of arranging a set of data in ascending or descending order.</p><h2> How do you calculate the CDF for i.i.d. random variables?</h2><p>The CDF (cumulative distribution function) for i.i.d. (independent and identically distributed) random variables can be calculated by first arranging the sample data in ascending order. Then, for each value in the sample, the CDF is equal to the proportion of values that are less than or equal to that value. This can be expressed as a formula: CDF(x) = (number of values ≤ x) / (total number of values).</p><h2> What is the significance of order statistics in statistics?</h2><p>Order statistics play a crucial role in statistics as they provide valuable information about the distribution of a population. They can be used to estimate parameters such as the mean, median, and standard deviation, and they also help in identifying outliers and extreme values in a dataset.</p><h2> Can order statistics be used for non-i.i.d. random variables?</h2><p>Yes, order statistics can also be used for non-i.i.d. random variables. However, the calculation of the CDF may be more complex as the variables are not independent and may have different distributions. In such cases, specialized techniques may be required to calculate the CDF.</p><h2> How can order statistics be applied in real-life scenarios?</h2><p>Order statistics have various applications in fields such as economics, finance, and engineering. In economics, they can be used to study income inequality, while in finance, they can help in analyzing stock market trends. In engineering, order statistics can be used to determine the strength of materials and predict failures in structures.</p>

FAQ: Order Statistics: CDF Calculation for i.i.d. Random Variables

What is the definition of order statistics?

Order statistics refers to the collection of random variables that represent the ordered values of a sample from a population. In other words, it is the process of arranging a set of data in ascending or descending order.

How do you calculate the CDF for i.i.d. random variables?

The CDF (cumulative distribution function) for i.i.d. (independent and identically distributed) random variables can be calculated by first arranging the sample data in ascending order. Then, for each value in the sample, the CDF is equal to the proportion of values that are less than or equal to that value. This can be expressed as a formula: CDF(x) = (number of values ≤ x) / (total number of values).

What is the significance of order statistics in statistics?

Order statistics play a crucial role in statistics as they provide valuable information about the distribution of a population. They can be used to estimate parameters such as the mean, median, and standard deviation, and they also help in identifying outliers and extreme values in a dataset.

Can order statistics be used for non-i.i.d. random variables?

Yes, order statistics can also be used for non-i.i.d. random variables. However, the calculation of the CDF may be more complex as the variables are not independent and may have different distributions. In such cases, specialized techniques may be required to calculate the CDF.

How can order statistics be applied in real-life scenarios?

Order statistics have various applications in fields such as economics, finance, and engineering. In economics, they can be used to study income inequality, while in finance, they can help in analyzing stock market trends. In engineering, order statistics can be used to determine the strength of materials and predict failures in structures.

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