Parametric function or statistic?

In summary, the conversation discusses the difference between a statistic and the expectation of a random variable or function. The use of "E" in the notation can be confusing, as it can refer to either the expectation or a parametric function. The expectation is a function that gives the probability-weighted average outcome of a random variable or function, while a statistic involves only a sample of values. It is incorrect to take the expected value of a specific sample, as a sample is already a set of determined values. The notation "EP" refers to the P-measure of a set, and can be calculated using integrals for continuous cases and inner products for discrete cases.
  • #1
EvLer
458
0
Which one would you say this is :confused: :

E(|x1-x2|)
x1, x2, xn - a sample of n values on the underlying random variable...
I was thinking this is a statistic :frown:
 
Physics news on Phys.org
  • #2
"E" of something is a probability-weighted average outcome, so it involves the entire set of outcomes rather than just a sample. A statistic involves only a sample. The x's in E[f(X1,x2)] are not sample values, they are shorthands for two random variables or distributions (e.g., "Normal distribution 1" and "Normal distribution 2").
 
  • #3
so... parametric function?
 
  • #4
"Probabilistic representation of the set of outcomes." I guess para. func. is acceptable for cases where the prob. rep. reduces to a parametric function.
 
  • #5
so, then what about |x1-x2|? would you say it is a statistic?
 
  • #6
Yes, that's a statistic. The initial confusing factor was your use of "E". Did you intend that to be the expectation? If so, E(u) is the expected value of a random variable or function of a random variable. It doesn't make sense to talk about the "expected" value of specific values from a sample.
 
  • #7
thanks a lot for explaining.
so to make sure I understand, it would be wrong to "do" expected value on a sample? since E's domain is data from entire distribution?
edit: would you call E() a function? if not, what is it formally? sorry if I don't get this right away...
Thank you again.
 
  • #8
Let c be a (deterministic) constant. Then the expected value of c is always c.

You can take the expectation of a sample, but the result will be the sample itself. That's because a sample {x1, ..., xn} (differently from random vars. {X1, ..., Xn}) is a set of values that have already been determined (by the very act of selecting them from a set of probable outcomes). There is nothing inherently probabilistic or random about a sample, once you have the sample.

And yes, in "standard" theory and applications, E[f(O)] (where "E" is a shorthand for "EP") is a function given by the integral of f(O)dP(O) where f is any (continuous, to be safe) function defined over the set of outcomes O, and P is the cumulative probability distribution defined over O. For discrete cases, it is the inner product f(O).p(O) where p is the density defined over O.

E[f(O)] is the P-measure of the set f(O), and hence the technically correct notation "EP."
 
Last edited:
  • #9
Examples:
Continuous P, specific parameters: Let U[0,1] be the uniform prob. dist. over the unit interval I=[0,1]: U[0,1](v) = v for v in I. Then E[U[0,1]] = EU[0,1][I] = [itex]\int_0^1 v dU[0,1](v) = \int_0^1 v dv = \frac{v^2}{2}|_0^1 = 1/2[/itex].
Continuous P, general parameters: Let U[a,b] be the uniform prob. dist. over any interval J = [a,b]: U[a,b](v) = (v-a)/(b-a) for v in J. Then
[tex]
E[U[a,b]] = E_{U[a,b]}[J] = \int_a^b v dU[a,b](v)
= \int_a^b \frac v{b-a} dv
=\frac{v^2}{2(b-a)}|_a^b = \frac{b^2-a^2}{2(b-a)} = \frac{a+b}2
[/tex]

Discrete P, specific params.: Let p be the uniform density function p(v) = 1/2 for all v in O = {0,1}. Then EP[{0,1}] = (1/2,1/2}.{0,1} = (1/2) 1 + (1/2) 0 = 1/2.
Discrete P, gen. params.: Let p be the uniform density function p(v) = 1/n for all v in W = {1, ..., n}. Then EP[W] = (1/n, ..., 1/n}.{1, ..., n} = (1 + ... + n)/n = n(n+1)/(2n) = (n+1)/2.
 
Last edited:

1. What is a parametric function?

A parametric function is a mathematical function that uses one or more parameters to define its shape or behavior. These parameters can be used to adjust or control the values of the function, allowing for flexibility and customization.

2. What is a parametric statistic?

A parametric statistic is a statistical method that assumes the data follows a specific distribution, such as a normal distribution. This allows for the use of mathematical models and equations to analyze the data and make inferences about the population.

3. What is the difference between a parametric and non-parametric function/statistic?

The main difference between parametric and non-parametric functions/statistics is that parametric methods assume a specific distribution for the data, while non-parametric methods do not make any assumptions about the underlying distribution. Parametric methods are often more powerful and efficient, but may not be suitable for all types of data.

4. How are parametric functions/statistics used in scientific research?

Parametric functions/statistics are commonly used in scientific research to analyze and interpret data. They can be used to test hypotheses, make predictions, and identify relationships between variables. They are also used to compare groups and determine the significance of results.

5. What are some common examples of parametric functions/statistics?

Some common examples of parametric functions/statistics include t-tests, ANOVA, linear regression, and chi-square tests. These methods are frequently used in fields such as biology, psychology, economics, and more to analyze and interpret data in a quantitative manner.

Similar threads

  • Set Theory, Logic, Probability, Statistics
2
Replies
54
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
335
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
302
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
280
  • Programming and Computer Science
Replies
14
Views
460
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
30
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
731
Back
Top