Calculating Certain Properties from Distribution Functions

[MathIsFun]

1. The problem statement, all variables, and given/known data
I am given a distribution function $f(x)$ that tells me the number of objects with a certain physical property $x$ (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 $x$ that satisfy a certain value of $f(x)$.

Homework Equations

I don't know

The Attempt at a Solution

If the function $f(x)$ is defined on $a<x<b$, I believe the total number of objects would be $$\int_{a}^{b}f(x)\,dx$$ and the average value of the property would be $$\frac{\int_{a}^{b}x f(x)\,dx}{\int_{a}^{b}f(x)\,dx}$$
First, are these correct?

Second, my main concern is that sometimes the property $x$ is defined only for values of $x$ in a set $A$ (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 $x$ that is held by $n$ objects and I get some value $x=m\notin A$, would I approximate it to the closest value that is in $A$, leave the answer as $x=m$, or say that there is no value of $x\in A$ that satisfies this condition?

Thank you

 

[kuruman]

First, are these correct?

They are correct.
The Census Bureau reports that the average American family in 2016 consists of 3.14 (no relation to π) persons. Does this answer your other question?

 

[MathIsFun]

Yes, thank you.

 

[MathIsFun]

Now I have to calculate the median of the distribution. If $f(x)$ is defined for $a<x<b$, I would then calculate $$h(x)=\int_{a}^{x} \frac{f(z)}{N}\,dz$$, where $N=\int_{a}^{b} f(x)\,dx$, to get the cumulative distribution $h(x)$ and solve for $x$ when $h(x)=0.5$.
Is this correct?

 

[kuruman]

That is correct. For future reference, if you have a second question after the first one has been answered to your satisfaction, please post it separately. You may also post a multipart question, as in parts (a), (b), etc. if you plan ahead.