cash king
New Member
Can anyone shed lights on why normal distribution only has the 1st and 2nd moments, while no other higher order moments (such as skewness and kurtosis)?
moments of normal distribution
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I may be slightly non-technical here but I'll give it a try. The normal distribution DOES have higher order moments. Indeed, it has an infinite number of moments, as do (most) continuous distributions. The unique characteristic of the normal distribution, although not unique to the normal only, is the fact that all even moments above the second moment can be written as functions of the second moment, for instance, the 4th moment is the 2nd moment squared. Regarding odd moments above the 2nd moment, all have a value of 0 since the distribution is symmetric. Thus, only the first two moments are relevant. Like I said, not very technical, but I think it addresses your question.
One more point of clarification. The normal distribution does have skewness and kurtosis. The skewness is 0 and the kurtosis is 3, which implies that the normal distribution has an excess kurtosis of 0.
Best
Brian
One more point of clarification. The normal distribution does have skewness and kurtosis. The skewness is 0 and the kurtosis is 3, which implies that the normal distribution has an excess kurtosis of 0.
Best
Brian
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cash king
New Member
thanks. my text book seems so confusing that your 'non-technicial' note really helps
Hihi, I am having some trouble deriving the Skew & Kurtosis of this question - can be found Reading 10 pg 46.
There is no mention of the distribution? Do we assume it is ~N(0,1)?
5. A bond has a default probability of 5%. Which is nearest, respectively, to the skew (S) and kurtosis (K) of the distribution?
Appreciate if anyone can shade some light on this.
There is no mention of the distribution? Do we assume it is ~N(0,1)?
5. A bond has a default probability of 5%. Which is nearest, respectively, to the skew (S) and kurtosis (K) of the distribution?
Appreciate if anyone can shade some light on this.
Hi @tosuhn
Source Q&A is here @ https://forum.bionicturtle.com/threads/p1-t2-307-skew-and-kurtosis-miller.6825/
When a question says, "A bond has a default probability of 5.0%" without further clarification, the unstated assumption is that the distribution is binomial with n=1 (for a single bond), or Bernoulli: there are only two outcomes, the bond with either default with probability of 5%, or survive with probability of 95%. Skew and Kurtosis are then calculated accordingly:
Source Q&A is here @ https://forum.bionicturtle.com/threads/p1-t2-307-skew-and-kurtosis-miller.6825/
When a question says, "A bond has a default probability of 5.0%" without further clarification, the unstated assumption is that the distribution is binomial with n=1 (for a single bond), or Bernoulli: there are only two outcomes, the bond with either default with probability of 5%, or survive with probability of 95%. Skew and Kurtosis are then calculated accordingly:
- skew = 3rd central moment/[StdDev]^3, and
- kurt = 4th central moment/[StdDev]^4, where the variance = n*p*(1-p), which in this case of one bond is simply 5%*95%, such that StdDev = sqrt(5%*95%). Thanks,
Thanks @David Harper CFA FRM CIPM
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