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I must confess I'm not an educator, but I like this question and at the very least I can answer with the intuitive picture I use in my own head. The $n$-th central moment $\mu_n$ of a random variable $X$ is defined by: $$\mu_n \equiv E[(X-E[X])^n]$$ For what follows we will take the measure space of $X$ to be the real numbers, since that's the easiest to ...

0

I think that the most interesting application of (sum of numerators/sum of denominators) “addition” is to continued fractions. For example, if you want to calculate the continued fraction expansion of the square root of 2,start with 1/0 and 0/1 and “add” them in the following way: In the top row you put the results of the “adding” that are greater then the ...

2

The problem is with the question, not with the students' answers. The question is ambiguous and I think the students' answer is actually much better than yours. Suppose I drive a thousand miles at 25mpg and you drive one mile at 35mpg. What's the average fuel efficiency? Your answer is 30mpg but I honestly can't think of any situation in which that is a ...

2

Think of an example with two ratios: 1/3 and 4/5. When you add the numerators, and divide this by the sum of the denominators, you get (1 + 4)/(3 + 5) = 5/8. Now, think about what is happening with the denominators - the denominator of the first ratio should only act on the first numerator. But instead, when you add the ratios in this way, the denominator ...

2

Allow me to offer another example: Imagine you and your best friend both want to buy a new smart phone. The phone you have chosen will cost you 300€ but your friend chooses a phone that will cost as much as 600€! Luckily, you have two vouchers that will give you a discount: The first voucher will give you the cheaper phone for free, if you buy two phones. ...

17

One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of numerators to sum of denominators is $\frac12$. However, we can also write $a=\frac03$ and $b=\frac22$. Now the ratio is $\frac25$, which is not equal to $\... 12 I like guest's answer. To elaborate, here is a possible question to ask them. You take two trips in your car: Trip 1 is a 100 mile drive that takes you 2 hours. Trip 2 is a 200 mile drive that takes you 1 hour. (a) What is the average speed of your car? (b) What is the average speed on an average trip? The answer to (a) is$\frac{...

13

It actually depends on exactly what you're asking. Or even what you SHOULD be asking. If you want the average profitability of all the 500+ operators in the Permian, you could just average all the profit margin percentages. This is taking the ratios (profit/revenue) for each company and averaging them. It corresponds to your expected (mean) profit margin ...

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