Quantitative Methods #6: Distributions

What is a probability distribution?

It is a list of all possible outcomes of an experiment that details the probabilities of each result. A distribution will either be discrete if the possible results are possible to count (example would be the number of days) or continuous for a distribution that has an infinite number of results. One example would be the height of someone. Why? Because you could always add more precision and in that regard, there is no end to the possible results. The distribution would be binomial if there are only two possible results.

Distributions in the CFA exam are generally assumed to follow a normal distribution, which you can see below:

For a normal distribution:

68% of results fall within +/- 1 standard deviation
90% of results fall within +/- 1.65 standard deviation
95% of results fall within +/- 1.96 standard deviation
99% of results fall within +/- 2.58 standard deviation

What you must remember is that with a confidence interval of:

-90% – You can assume that the result is within +/- 1.645 standard deviations
-95% – You can assume that the result is within +/- 1.96 standard deviations
-99% – You can assume that the result is within +/- 2.575 standard deviations

There is no logic but you should know these. The “standard” distribution has a mean of 0 and a standard deviation of 1. Obviously, if you get different “inputs” in a CFA exam, you must adjust.

If you want to get a reference as to how likely or unlikely a result is bound to happen, you can use tests such as:

Z-Test:

Then you can simply compare the Z-Test to the table and you will see the “probability.”

SFratio:

This ratio is used to determine the probability that a portfolio’s return will be below a given point (RL).

When you are testing a hypothesis, you start from a certain assumption H1 which is the null hypothesis. If you are able to reject that hypothesis with the given confidence level, you would be choosing the alternative hypothesis, Ha

You might obviously be making mistakes while doing so

Type 1 error: Reject hypothesis when it is actually true
Type 2 error: Do not reject it when it is false


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