# Probability and Estimation Question

This is from the CFAI Text. I don’t understand how the answer was reached. Any insight would be appreciated. An exchange rate has a given expected future value and standard deviation. A. Assuming that the exchange rate is normally distributed, what are the probabilities that the exchange rate will be at least 1, 2, or 3 standard deviations away from its mean. B. Assume that you do not know the distribution of exchange rates. Use Chebsyshev’s inequality (that at least 1 – 1/k2 proportions of the observations will be within k standard deviations of the mean for any positive integer greater than 1) to calculate the maximum probability that the exchange rate will be at least 1, 2, or 3 standard deviations away from its mean. Answers: A: • P(|X - µ| >= 1σ) = 0.3174 • P(|X - µ| >= 2óσ) = 0.0456 • P(|X - µ| >= 3σ) = 0.0026 B: • P(|X - µ| >= 1σ) <= (1/1)^2 = 1 • P(|X - µ| >= 2σ) <= (1/2)^2 = 0.25 • P(|X - µ| >= 3σ) <= (1/3)^3 = 0.1111

Part A just wants you to lookup 1 in the cumulative z-table. For 1 stdev, this gives you .8413, and is P((X-mu)/sigma < 1). Therefore P((X-mu)/sigma > 1) = 1-.8413 =0.1587 because you want P( (X-mu)/sigma > 1 or < -1) and the normal distribution is symmetrical, you get: P(|X - µ| >= 1σ) = 2*.1587=0.3174 What don’t you understand about Chebyshev?

Edit: Forget it

A technical note here, from the CFAI text, “Chebyshev’s inequality states that for any set of observations, the proportion of the observations within k standard deviations of the mean is at least 1 - 1/k2 for all k > 1” for the case k=1, does this inequality still hold?

If k =1 that statement reads “Chebyshev’s inequality states that for any set of observations, the proportion of the observations within k standard deviations of the mean is at least 0” so yes it holds. Also, k does not have to be an integer for that to hold.

>B: >• P(|X - µ| >= 1σ) <= (1/1)^2 = 1 >• P(|X - µ| >= 2σ) <= (1/2)^2 = 0.25 >• P(|X - µ| >= 3σ) <= (1/3)^3 = 0.1111 just noticed, in the last line you have (1/3)^3. Should be (1/3)^2, although you have the right answer (1/9).

Could you please explain once again why did you multiply 0.1587 by 2?

One 0.1587 is above the mean, the other is below the mean.