Order statistics and a convolution

In summary, the conversation is about a problem involving random samples from a uniform distribution and finding the probability density function (pdf) and cumulative distribution function (cdf) for the largest and smallest numbers in the sample. The conversation also discusses finding the expected value and limit of the difference between the largest and smallest numbers. The conversation then moves on to finding the pdf for Y2 and Y, with confusion about calculating the integral.
  • #1
strangequark
38
0

Homework Statement



Hi, I'm having some problem with one of my final exam study questions, and I'm hoping someone can help me out a little.

here is the problem:

Let [tex]Y_{1},Y_{2},...,Y_{n} [/tex] denote random samples of numbers from a uniform distribution on the interval [0,1]. Denote the largest and smallest numbers as [tex]Y_{n}'[/tex] and [tex]Y_{1}'[/tex].

a. Find the pdf and cdf of [tex]Y_{1}'[/tex] and[tex]Y_{n}'[/tex].
b. Find [tex]E(Y_{n}'-Y_{1}')[/tex]
c. Show that [tex] \lim_{n\to\infty}E(Y_{n}'-Y_{1}')=1[/tex]
d. Find the pdf for [tex]Y_{2}[/tex] and find the pdf for [tex]Y=Y_{1}+Y_{2}[/tex]

Homework Equations



the ith order statistic:

[tex]f_{Y_i}=\frac{n!}{(i-1)!(n-i)!}(F_{Y}(y))^{i-1}(1-F_{Y})^{n-i}f_{Y}(y)[/tex]

convolution:
if [tex]X=Y_{1}+Y_{2}[/tex]

[tex]f_{X}(x)=\int^{x}_{0}f_{Y_{1}}(y)f_{Y_{2}}(x-y)dy[/tex]

The Attempt at a Solution



i have:
a.

ok, so I am pretty sure that my pdf's will be:
[tex]f_{Y_{1}'}=\frac{n!}{(n-1)!}(1-y)^{n-1}=n(1-y)^{n-1}[/tex]
and
[tex]f_{Y_{n}'}=\frac{n!}{(n-1)!}(y)^{n-1}=ny^{n-1}[/tex]

b. cumulative functions easy from above.

c.
I get
[tex]E(Y_{n}')=\frac{n}{n+1}[/tex]
and
[tex]E(Y_{1}')=\frac{1}{n+1}[/tex]
so:
[tex]E(Y_{n}'-Y_{1}')=\frac{n-1}{n+1}[/tex]

d. here's where I'm confused... for [tex]f_{Y_{1}}[/tex] and [tex]f_{Y_{2}}[/tex], I have:

[tex]f_{Y_{1}}=n(1-y)^{n-1}[/tex]

Is my [tex]f_{Y_{2}}[/tex] correct?

[tex]f_{Y_{2}}=n(n-1)y(1-y)^{n-2}[/tex]

if it is, how the heck do I calculate this integral??

[tex]\int^{x}_{0}n(n)(n-1)(1-y)^{n-1}(x-y)(1-(x-y))^{n-2}dy[/tex]

I'm not seeing my mistake. The answer should be:

[tex]f_{X}(x)=1-|1-x|[/tex] for [tex]0\leqx\leq2[/tex]

help?
 
Last edited:
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  • #2
[itex]f_{Y_{2}}(z)[/itex] is n!/((n-k)!(k-1)!)F(z)k-1[(1-F(z)]n-kf(z) = n(n-1)y(1-y)n-2, so it is correct.
 

1. What is the definition of order statistics?

Order statistics refers to a set of statistical methods used to analyze and interpret a collection of ordered data. It involves finding and analyzing various statistics, such as the minimum, maximum, median, and quartiles, among others, from a set of data arranged in ascending or descending order.

2. How is the concept of order statistics applied in real-world scenarios?

Order statistics is commonly used in fields such as economics, finance, and engineering to analyze data sets and make informed decisions. For example, in finance, order statistics can be used to determine the best and worst performing stocks in a portfolio. In engineering, it can be used to identify the weakest and strongest components in a system.

3. What is the purpose of using a convolution in order statistics?

A convolution is a mathematical operation that involves combining two functions to produce a third function. In the context of order statistics, a convolution is used to determine the probability distribution of the sum of two or more random variables. This is useful in analyzing data sets with multiple variables, such as in finance where the returns of different stocks are combined.

4. Can order statistics and a convolution be used together to analyze non-numerical data?

Yes, order statistics and a convolution can be applied to analyze non-numerical data, such as rankings or ratings. In this case, the data can be converted to numerical values and then analyzed using order statistics and a convolution. This approach is commonly used in social sciences to analyze survey data.

5. Are there any limitations to using order statistics and a convolution?

One limitation of using order statistics and a convolution is that they assume the data is independent and identically distributed (iid). This means that the data should be randomly sampled and follow the same underlying distribution. If these assumptions are not met, the results of the analysis may not be accurate. Additionally, these methods may be computationally intensive for large data sets.

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