Using Multiple integrals to compute expected value

[WMDhamnekar]

Homework Statement

If you were to pick n > 2 random real numbers from the interval (0,1), what are the expected values for the smallest and largest of those numbers?

Relevant Equations

None

I want to know how did author derive the red underlined term in the following Example?

 

[PeroK]

First, I'm not sure why the author invokes the complexity of a joint pdf. Try working on $X$ and $Y$ separately. It is much simpler.

The basic idea is that if $X = x$ and $Y = y$, then all $n$ variables must be in the range $y-x$.

 

[WWGD]

Since the two RVs are independent, the computation should be straighforward. Otherwise, it can't be known without additional information. Edit. Please see my post #5 below. They are dependent.

 

[Orodruin]

Since the two RVs are independent, the computation should be straighforward. Otherwise, it can't be known without additional information.

X and Y are certainly not independent.

 

[WWGD]

My bad, I misread. Clearly Max> Min, so once y is selected, need X>Y.

 

[Orodruin]

How so? How does knowing one change or affect the other . More formally:
Isn't $P(M=Max| m=Min) \neq P(M=max)$?

No. For example, $\lim_{x\to 0} E(Y|X=x) = (n-1)/n \neq n/(n+1) = E(Y)$. It should also be clear from the form of $f(x,y)$ that $f(x,y) \neq f_X(x) f_Y(y)$.

However, all you need to know is that the $U_i$ are independent and uniformly distributed in (0,1).

 

[WWGD]

No. For example, $\lim_{x\to 0} E(Y|X=x) = (n-1)/n \neq n/(n+1) = E(Y)$. It should also be clear from the form of $f(x,y)$ that $f(x,y) \neq f_X(x) f_Y(y)$.

However, all you need to know is that the $U_i$ are independent and uniformly distributed in (0,1).

Yes, I rewrote and corrected in #5.

 

[WMDhamnekar]

No. For example, $\lim_{x\to 0} E(Y|X=x) = (n-1)/n \neq n/(n+1) = E(Y)$. It should also be clear from the form of $f(x,y)$ that $f(x,y) \neq f_X(x) f_Y(y)$.

However, all you need to know is that the $U_i$ are independent and uniformly distributed in (0,1).

Following reading material is necessary to answer my questions. After reading the following material, you may answer my following questions and therby clear my doubts:
1) What is the meaning of 1 ?

2) How did author make 2 = $n (n-1) (y-x)^{n-2}$

3) How to prove 3 ?
Order Statistics:

Kth Order Statistics:

 

[Orodruin]

1) Which part of (1) are you having trouble with? $(-\infty,x]$ and $(x,\infty)$ are intervals on the real line.

2) By inserting the distribution function for the random variables. For a stochastic variable uniformly distributed on (0,1) $f(x) = 1$ on that interval and consequently $F(x) = x$.

3) Think about what that statement means in terms of the variables.

 

[WMDhamnekar]

1) Which part of (1) are you having trouble with? $(-\infty,x]$ and $(x,\infty)$ are intervals on the real line.

2) By inserting the distribution function for the random variables. For a stochastic variable uniformly distributed on (0,1) $f(x) = 1$ on that interval and consequently $F(x) = x$.

3) Think about what that statement means in terms of the variables.

What is x? Why are its lower and upper limits $-\infty$ and $+ \infty$ respectively?

Does x(any real number) relate to the size of n parts manufactured in the single run of assembly line?

 

[Orodruin]

What is x? Why are its lower and upper limits −∞ and +∞ respectively?

x is any real number.

The rest of your post is unclear to me in termms of what you are actually asking. Please keep your questions contained without branching out in all kinds of directions. This will help focus on particular misunderstandings and issues.

 

[WWGD]

What is x? Why are its lower and upper limits $-\infty$ and $+ \infty$ respectively?

Does x(any real number) relate to the size of n parts manufactured in the single run of assembly line?

x is the length of a manufactured part.

 

[WMDhamnekar]

How to write a computer program in any language (octave, Java, C,C++) that verifies the results in this example for the case n =3 and n=4 by taking a large number of samples?

 

[PeroK]

How to write a computer program in any language (octave, Java, C,C++) that verifies the results in this example for the case n =3 and n=4 by taking a large number of samples?

What do you mean by "how to write a computer program"? If you know a programming language like Python it's just a standard programming exercise.

 

[WWGD]

How to write a computer program in any language (octave, Java, C,C++) that verifies the results in this example for the case n =3 and n=4 by taking a large number of samples?

Just generate several random triplets from within (0,1) and order them from smaller to larger. Find their mean, and plot them ( All Minima, All medium, All Maxima) test against the respective distributions of each of the Order Statistics. Note this is not a proof of any sort, but a way of testing a model.

 

[WWGD]

How to write a computer program in any language (octave, Java, C,C++) that verifies the results in this example for the case n =3 and n=4 by taking a large number of samples?

For more details, context, I suggest John Mandel's " The Statistical Analysis of Experimental Data". Not a textbook , but a pretty interesting read. Mandel worked for the National Bureau of Standards, way back in the 50's-70's. Specially if you're interested in the area and/or will be using it for your major. Some applications: You study floods, to estimate the Max and Min volume of floods in a given area , in order to have a reasonable margin of error (and not to overdo it when building dams). IIRC, you can also use them as tests of fit for a given distribution: The distribution of Order Statistics associated to specific distributions are pretty different from each other, so that you can test whether a sample $(X_1, X_2,...,X_n)$ comes from a given distribution.

 

[haruspex]

"pick n > 2 random real numbers from the interval (0,1)"

Just to point out that the author has forgotten to specify that the picks are not only independent but, specifically, using a uniform distribution on the interval. "Random", by itself, tells you nothing; a deterministic selection is just a special case of random.