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Suppose we have the following three curves and we know that one of them is normal. How can we prove that the other two curves cannot be normal? (We don't have to determine which one of the three curves is normal)

Are there any theorems that normal distributions follow that other bell-shaped curves don't? How can one sho, preferably rigorously, that two of the three curves below cannot be normal given that one of the curves is normal?

enter image description here

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    $\begingroup$ In this particular case, all three shapes look normal shaped and they seem to stretch to match each other, so you are just left with the area/total probability test. You do not have a vertical scale, so deciding which may be difficult. There are other bell shaped curves which would not match and you could demonstrate this graphically or numerically $\endgroup$
    – Henry
    Commented Nov 30, 2020 at 1:55
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    $\begingroup$ How do you generate those curves? $\text{//}$ If your “density” doesn’t integrate to $1$, it isn’t a density, so I do not follow what you mean. $\endgroup$
    – Dave
    Commented Nov 30, 2020 at 1:58
  • $\begingroup$ @Dave I found these curves in a textbook. I think my idea about the area under the curves not being 1 was incorrect. I'll edit my question. $\endgroup$
    – Ricky_Nelson
    Commented Nov 30, 2020 at 2:00
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    $\begingroup$ As you said, the one with a total probability (i.e. area under the density curve) of $1$ is a probability density curve. Those with a different area under the curve are not. Visually the black/solid curve has the greatest area, and then I think the green/dotted curve, followed by the red/dashed curve. So the areas are different, and at most one can be $1$. $\endgroup$
    – Henry
    Commented Nov 30, 2020 at 2:11
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    $\begingroup$ There isn't anything specific to the normal distribution about the textbook question. The question could have been "explain why at most one of these three curves is a probability density function" and the answer would have been the same. $\endgroup$
    – fblundun
    Commented Nov 30, 2020 at 10:05

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Here is a path as to how to visually determine whether bell-shaped curves are normal.

Start by drawing equal probability mass cut points on the graph. This produces quantile (cut point) estimates. Per a source, to quote:

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created.

Next, examine the cut-points and visually (or more rigorously) construct a QQ Plot. Per an educational source:

One way to assess how well a particular theoretical model describes a data distribution is to plot data quantiles against theoretical quantiles...For a large sample from the theoretical distribution the plot should be a straight line through the origin with slope 1...

where the theoretical distribution here is taken to be the Normal distribution.

To obtain a probability that the slope estimate per a Least-Squares fit is indeed here equal to 1, more work can be performed as detailed in this source: Non-Zero Null Tests for Simple Linear Regression.

Optionally, I suggest you actually simulate a Normal and other distributions based on the sample size used to construct the graph, and test for accuracy over repeated runs the suggested methodology.

I hope this answer is of assistance.

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  • $\begingroup$ -1 This answer misses the point of the question, which is elaborated in comments to the question. $\endgroup$
    – whuber ♦
    Commented Nov 30, 2020 at 14:47
  • $\begingroup$ Whuber: Yes, I did want to avoid comment on a fundamental mischaracterization of a potentially interesting problem. In essence, I am assuming the point of the comments are to state that for curves to be truly representative of potential distributions, they all must (obviously) sum to unity. That being finally agreed upon, how could one ever even proceed? To that end, assuming the question is edited, I have presented a workable testable valid methodology. I would be interested if you have any conceivable alternative paths in mind? $\endgroup$
    – AJKOER
    Commented Nov 30, 2020 at 17:37
  • $\begingroup$ At the very beginning, you ask the viewer to perform a nearly impossible task: namely, to slice the area under the curve into vertical strips having equal areas. Very few people will be able to do that with any accuracy. At that point we have gone so far astray from the essential point of the question that it's not suitable to discuss any further steps. $\endgroup$
    – whuber ♦
    Commented Nov 30, 2020 at 17:46
  • $\begingroup$ @whuber: Yes, difficult perhaps, but not impossible. For example, draw line segments over the curve, and use geometry to estimate sector areas. Total the area estimates and divide by a select # of proposed cuts. Per the geometry exercise, estimate where such approximate equal area sectors cuts would be located. Tedious, but perhaps a quantitatively rewarding/revealing exercise nevertheless. $\endgroup$
    – AJKOER
    Commented Dec 9, 2020 at 0:50
  • $\begingroup$ Now, if someone would suggest how large a # of random say normal (or other) deviates to generate, I may find time to run a few simulations, then generate the graphs and perform various linear sector examinations and subsequent quantile estimation. Knowing the true distribution and parameters used to perform the simulation may allow me to assess general accuracy and possible feasibility. Preferably, someone else should perform this exercise (credibility). No assertion to have completely developed all the best practices here to necessarily best address this distribution ID exploratory study. $\endgroup$
    – AJKOER
    Commented Dec 9, 2020 at 1:59

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