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Errors Found in Study Materials P1.T2. Quantitative Methods

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @Mdeclercq Apologies for the confusing statement: I think that's a legacy statement from an FRM syllabus source that does need to be corrected (I have made a note, hence your placement here in Errors is appropriate). I haven't seen this paper, but I completely agree that the most accurate definition of kurtosis refers to tail density (as I have written in this forum many times; e.g., https://www.bionicturtle.com/forum/threads/kurtosis-and-peakedness.4758/post-12546).

Here is a good thread https://www.bionicturtle.com/forum/threads/error-in-millers-illustration-of-leptokurtosis.7685/ e.g.,
HI @brianhfield

Love you diagram! :cool:

I agree that Miller's plot choice is unfortunate, but authors occasionally do select this rendering (i.e., where the student's t appears less peaked). As @irenab writes, implicitly he is rescaling neither the normal nor the student's t (although Miller's plots are stylistic in the first place: there are not y pdf values). Without any adjustment, the student's t will appear less peaked, but at the same time, it will have a variance = df/(df-2) > 1. If normal the normal is rescaled to match the variance (an apples-to-apples for the second moment, if you will), the expected higher peakedness of the student's t will be revealed; i.e., if the variances match, the comparison will look like your diagram!

Okay, but it turns out that kurtosis does not (100%) correspond to both higher peaks and heavier tails necessarily; rather, it is just the majority use case and the intuitive expectation. See http://stats.stackexchange.com/questions/80626/kurtosis-of-made-up-distribution

Here is a recommended paper, "On the Meaning and Use of Kurtosis"

For example,
Why are tailedness and peakedness both components of kurtosis? It is basically because kurtosis represents a movement of mass that does not affect the variance. Consider the case of positive kurtosis, where heavier tails are often accompanied by a higher peak. Note that if mass is simply moved from the shoulders of a distribution to its tails, then the variance will also be larger. To leave the variance unchanged, one must also move mass from the shoulders to the center, which gives a compensating decrease in the variance and a peak. For negative kurtosis, the variance will be unchanged if mass is moved from the tails and center of the distribution to its shoulders, thus resulting in light tails and flatness."

For several reasons including technical accuracy, my preferred short description is simply that leptokurtosis signifies "heavy tails" (a runner-up would be "fat tails," but this can also lead to subtle misunderstandings vis a vis thin versus fat. We will fix this on the next revision. Thank you!

btw, here is a recent video of mine on kurtosis where I say that while kurtosis tends to associate with peakedness (i explain why this is the case and, in my experience, it is the tendency) the better definition of leptokurtosis is heavy tails
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Nicole Seaman

Chief Admin Officer
Staff member
Thread starter #44
In page 45 of R14-P1-T2, the formula seems not right. There's no variable Expn in it.
Hello @tattoo

It helps if you provide detailed information about which formula you are referring to. I believe this is the formula you are referencing but I want to make sure. This helps us so we don't have to go to the source to look for the formula that you are referring to, and it saves David and others time since they can most likely answer your question just by looking at the formula and not having to search through the notes.


Thank you,



New Member
Hi David. In the study notes for "Miller, Mathematics & Statistics for Financial Risk Management, Chapters 2,3, 4, 6 and 7", I found a small typo.
On Page 28 in the standard deviation formula: In the term to the right, there is the index i, but the sum over i's is missing.


New Member
Hey, did you remove the study notes for Miller, Mathematics & Statistics? As you can see in the latest post I was working through them on Monday, just to see they apparently have disappeared. Will they be reuploaded or are you completely removing them?

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @czet Yes, of course they will be re-uploaded, but the exam was just administered and we are going through everything to update for 2020, as usual. There is always some such period at this point in the year when the syllabus changes. Thank you!
Good Day David,

An excerpt from the study notes say this:


Instead i would say: [ w\wedge2\sigma a\wedge2\;+\;(1-w)\wedge2\sigma b\wedge2\;+\;2.w.(1-w).\sigma ab \]

That should be more accurate.
Hi David,

Kindly correct/clarify...(from QA-6 study notes-page 16)

"Because the null hypothesis is either true or false, and we make a binary decision, we can commit one of two errors. To mistakenly reject a true null hypothesis is to commit a Type I error. We denote this Prob [Type I error | true null] with an alpha, α, and it is also called the test “size.” The most common test sizes are 1.0% and 5.0%.

If we fail to reject a true null, then we commit a Type II error. We denote this
Prob [Type II error | false null] with beta, β. The value (1 - β) is called the power of the test; therefore, the power is the probability that a false null is correctly rejected."

A type II error is committed when we fail to reject a true false null hypothesis
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New Member
Hi David and Nicole,

When going through page 15 of T2- Chapter 2 - Random Variables

I found some typos in the attachment:
PMF is for discrete random variable
PDF is for continuous r.v.
Also, the denominator in the power of natural e is missing a 2 in the normal distribution.



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