- #1
MathIsFun
1. The problem statement, all variables, and given/known data
I am given a distribution function [itex]f(x)[/itex] that tells me the number of objects with a certain physical property [itex]x[/itex] (such as having a certain mass or temperature) and I need to calculate the total number of objects, the average value of the property of the objects, and values of [itex]x[/itex] that satisfy a certain value of [itex]f(x)[/itex].
I don't know
If the function [itex]f(x)[/itex] is defined on [itex]a<x<b[/itex], I believe the total number of objects would be [tex]\int_{a}^{b}f(x)\,dx[/tex] and the average value of the property would be [tex]\frac{\int_{a}^{b}x f(x)\,dx}{\int_{a}^{b}f(x)\,dx}[/tex]
First, are these correct?
Second, my main concern is that sometimes the property [itex]x[/itex] is defined only for values of [itex]x[/itex] in a set [itex]A[/itex] (e.g., counting). Since the distribution function is an approximation, does it still work for these values? For instance, if I had to calculate the value of the property [itex]x[/itex] that is held by [itex]n[/itex] objects and I get some value [itex]x=m\notin A[/itex], would I approximate it to the closest value that is in [itex]A[/itex], leave the answer as [itex]x=m[/itex], or say that there is no value of [itex]x\in A[/itex] that satisfies this condition?
Thank you
I am given a distribution function [itex]f(x)[/itex] that tells me the number of objects with a certain physical property [itex]x[/itex] (such as having a certain mass or temperature) and I need to calculate the total number of objects, the average value of the property of the objects, and values of [itex]x[/itex] that satisfy a certain value of [itex]f(x)[/itex].
Homework Equations
I don't know
The Attempt at a Solution
If the function [itex]f(x)[/itex] is defined on [itex]a<x<b[/itex], I believe the total number of objects would be [tex]\int_{a}^{b}f(x)\,dx[/tex] and the average value of the property would be [tex]\frac{\int_{a}^{b}x f(x)\,dx}{\int_{a}^{b}f(x)\,dx}[/tex]
First, are these correct?
Second, my main concern is that sometimes the property [itex]x[/itex] is defined only for values of [itex]x[/itex] in a set [itex]A[/itex] (e.g., counting). Since the distribution function is an approximation, does it still work for these values? For instance, if I had to calculate the value of the property [itex]x[/itex] that is held by [itex]n[/itex] objects and I get some value [itex]x=m\notin A[/itex], would I approximate it to the closest value that is in [itex]A[/itex], leave the answer as [itex]x=m[/itex], or say that there is no value of [itex]x\in A[/itex] that satisfies this condition?
Thank you