# Homework Help: Stats: Biomdal Question

1. Jul 21, 2012

### BryMan92

1. The problem statement, all variables and given/known data

I am analyzing fuel economy of vehicles in Europe. Since there are gasoline (lower MPG) and diesel (higher MPG) I assumed the distribution to be bimodal. I am trying to deconstruct the data into a gasoline MPG and a diesel MPG. I have attached an image to explain my approach.

2. The attempt at a solution

I'm no statistician, so forgive me while I break math. I figured I could treat this system like balancing a lever on a fulcrum. Essentially, I said a diesel vehicle is 40% more than a petrol vehicle and the red line is a given average of the MPGs. So, I did the following maths:

% of Cars for Petrol * x = % of Diesel* (1-x)
x=7/10, x'=3/10

So, I said the gasoline vehicles would receive a 70% (the distance) *40% (difference in fuel economy) reduction, or (1-.7*.4)(Average), while diesel would receive a (1+30%*40%)(Average). Would this, by any chance, work or be accurate??

Thanks guys!!!!

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2. Jul 21, 2012

### chiro

Hey BryMan92 and welcome to the forums.

There is a technique known as the EM algorithm, and when you apply it to a mixture of normal distributions with their own peaks (i.e. in this case for diesel and other), you can calculate the PDF that converges to that particular distribution with the bimodal property.

From there you can calculate anything from expectation, variance, probabilities and so on.

3. Jul 21, 2012

### BryMan92

Thanks, chiro! I am glad to know there is a way to do this. My problem is, I do not have a large stats background. From what Google told me, it doesn't seem likely that I will be able to learn how to do this.

I am curious, would I even have enough information to solve it? I know % diesel, % petrol, and the estimated difference in MPG. Is there some kind of formula I could plug into?

Thank you so much!!

4. Jul 22, 2012

### awkward

On a more basic level, have you looked at the data? Is it, in fact, bimodal?

5. Jul 22, 2012

### BryMan92

I do not have the data, I just have an average of the ENTIRE MPG of the country. I know %diesels and %gas vary change, but other than exact #s, I have some data that shows the average of a gasoline is 48 MPG, while the average of the diesel is 69 MPG.

The difference in MPG between diesel and gasoline is pretty large, so I assume it would be bimodal. For some countries, the ratio between the amount of gasoline and diesel vehicles was close leading me to think the average is too high for gasoline and too low for diesel. This is why I assume bimodal. But, for countries that are 75/25 I just assumed the average was close enough to be treated as say a diesel estimate.

Image 1 shows equal amounts, while image 2 shows a bias.

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Last edited: Jul 22, 2012
6. Jul 22, 2012

### uart

Ok Bry, so what you're really asking is this:

- You have a fleet that consists of petrol vehicles P and diesel vehicles D.

- You know the percent which are P and the percent which are D. Say $N_P/N = \alpha$ and $N_D/N = (1-\alpha)$, where N is the size of the entire fleet.

- You know (estimate) that the average fuel efficiency of the diesel fleet is some given constant times that of the petrol fleet, $\bar{D} = k \bar{P}$.

- You know the average fuel economy (MPG) of the entire fleet, lets call it $\bar{X}$

- From the above data you wish to estimate the values of $\bar{P}$ and $\bar{D}$, the individual fuel economies of the petrol and diesel fleets.

Is that a correct problem statement? If so this is very easily solved.

7. Jul 22, 2012

### uart

... continued.

The overall mean is just a linear combination of the two individual means,

$$\bar{X} = \alpha \bar{P} + (1-\alpha)\bar{D}$$

Which gives,

$$\bar{X} = \alpha \bar{P} + k(1-\alpha)\bar{P}$$

$$\bar{X} = [\alpha + k(1-\alpha)] \bar{P}$$

Now rearranging for $\bar{P}$ gives,

$$\bar{P} = \frac{\bar{X}} {\alpha + k(1-\alpha)}$$

and therefore,

$$\bar{D} = \frac{k \bar{X}} {\alpha + k(1-\alpha)}$$

Last edited: Jul 22, 2012
8. Jul 22, 2012

### BryMan92

Uart, this is wicked awesome and exactly what I needed! Thank you so, so, so much!

I have a small amount of other, but I will neglect them! Thank you!!

Last edited: Jul 22, 2012
9. Jul 22, 2012

### Ibix

Just a (belated) observation on 'bimodal'. If I were you I'd hit some manufacturer websites and get an idea of the range of values. You might also try contacting your government department responsible for transport. Some are very helpful and have access to raw data that can give you good answers with no modelling.

10. Jul 22, 2012

### BryMan92

I do have a small set of data that compares 10 vehicles with petrol and diesel engines (from the makers website), and a large amount from a website called fuelly.com. I hope this image will demonstrate the biomodality!

Also, I am actually working with my countries Department of Energy, already!

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