Chebyshev’s Inequality

CFA Level I Quant
#1

How you`re supposed to know Chebyshev’s inequality table??

I mean see the exercise:

According to Exhibit 27, the arithmetic mean monthly return and standard deviation of monthly returns on the S&P 500 were 0.95 percent and 5.39 percent, respectively, during the 1926–2017 period, totaling 1,104 monthly observations. Using this information, address the following:

1 Calculate the endpoints of the interval that must contain at least 75 percent of monthly returns according to Chebyshev’s inequality.

How I`m supposed to know that 75 percent is 2 standard deviations??

There`s a table in page 421 that states the percentage or k values of 1,25, 1,5, 2, 2,5 3 4

But how I`m supposed to know how many standard deviation I should put?

WTF

#2

Chebyshev’s inequality says that:

P(|X – μ| / σ ≤ k) ≥ 1 – 1/k2

So, if they give you, say, k = 3, then,

P(|X – μ| / σ ≤ 3) ≥ 1 – 1/32 = 1 – 1/9 = 8/9 = 88.89%

If they give you, say, P(|X – μ| / σ ≤ k) ≥ 75%, then,

75% = 1 – 1/k2

1/k2 = 1 – 75% = 25% = ¼

k2 = 4

k = √4 = 2

Don’t memorize it; understand it.

#3

The proportion of the observations within k standard deviations of the arithmetic mean is at least 1 − 1/k2 for all k > 1.

The book didn`t showed the proportion in the formula ex= x% = 1 - 1/K2

I just saw the formula how was it, but the book actually says it but its was wrote before.

Thanks your explanation clarified it all!

#4

I looked at the 2019 curriculum and it didn’t have it written out as I did.

Stupid.

#5

Also when solving the book didn`t showed the use of this formula:

Solution to 1: According to Chebyshev’s inequality, at least 75 percent of the observations must lie within two standard deviations of the mean, X ± 2s. For the monthly S&P 500 return series, we have 0.95% ± 2(5.39%) = 0.95% ± 10.78%. Thus the lower endpoint of the interval that must contain at least 75 percent of the observations is 0.95% − 10.78% = −9.83%, and the upper endpoint is 0.95% + 10.78% = 11.73%.

It just assumed you knew it by the table, they should have showed the use of the formula you stated above.

#6

Yup.

Stupid.

#7

Exactly they messed up with that they should have showed that!