Contents
- 1 How is the sample range related to order statistics?
- 2 How are order statistics used in data analysis?
- 3 Which is the ith order statistic of the sample?
- 4 Is the nth order statistic the maximum or minimum?
- 5 Can a random variable arise from more than one population?
- 6 How are X and Y independent random variables?
The sample range is the difference between the maximum and minimum. It is a function of the order statistics: A similar important statistic in exploratory data analysis that is simply related to the order statistics is the sample interquartile range.
How are order statistics used in data analysis?
One of the first steps in exploratory data analysis is to order the data, so order statistics occur naturally. In particular, note that the extreme order statistics are x ( 1) = min {x1, x2…, xn}, x ( n) = max {x1, x2, …, xn} The sample range is r = x ( n) − x ( 1) and the sample midrange is r 2 = 1 2[x ( n) − x ( 1)].
Which is a special case of an order statistic?
Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.
Which is the ith order statistic of the sample?
where the subscript (i) enclosed in parentheses indicates the ith order statistic of the sample. The first order statistic (or smallest order statistic) is always the minimum of the sample, that is,
Is the nth order statistic the maximum or minimum?
Similarly, for a sample of size n, the nth order statistic (or largest order statistic) is the maximum, that is, The sample range is the difference between the maximum and minimum. It is a function of the order statistics:
How are order statistics used in probability theory?
Order statistic. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution .
Can a random variable arise from more than one population?
In general, the random variables X1., Xn can arise by sampling from more than one population. Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat–Beg theorem .
How are X and Y independent random variables?
Conversely, X and Y are independent random variables if for all x and y, their joint distribution function F(x, y) can be expressed as a prod- uct of a function of xalone and a function of yalone (which are the marginal distributions of andX Y, respec- tively).