QM PREREQ4 – Common Probability Distributions Flashcards

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1
Q

What is a probability distribution?

A

Specifies the probabilities associated with the possible outcomes of a random variable

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2
Q

What are the 7 common probability distributions and why are they useful to know?

A

Uniform, binomial, normal, lognormal, Student’s (named after a person called Student!), chi-square, or F-distribution

Most distributions will look like one of these 7
So when we see a distribution we can say it is an “approximately normal” or “approximately chi square” distribution
This is useful because each of these common distributions has well-defined mathematical properties, which we can then use to analyse and interpret our data

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3
Q

What is a random variable and what are the two forms it can take?

A

A random variable is a quantity whose future outcoems are uncertain.

It can be either
- Discrete: take on at most a countable number of possible values (possibly infinite)
- Continuous: cannot count the possible values

Every random variable is associated with a probability distribution that describes the variable completely

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4
Q

What is a probability function?

A

Specifies the probabilities that a random variable can take

For discrete variables we would use p(x)
For continuous variables we would use the probability density function

The probability function has two key properties.
1. 0 =< p(x) =< 1 (any given probability within the data must be between or equal to 0 and/or 1
2. sum p(x) over all values of x equals 1. That is, if you add up all the values beneath the probability function they should add to 1

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5
Q

What is a CDF?

A

Cumulative distribution function
Gives the probability that a variable X is less than equal to a particular value x
Can be used for percentile rank for example
It is a slope that goes from 0 to 1 (0 to 1 being on the y axis)

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6
Q

What is a discrete uniform distribution?

A

All outcomes are equally likely
The probability distribution is a rectangle
Thus length x width = 1

It will look like stairs of equal height and width as a CDF

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7
Q

What is a continuous uniform distribution?

A

The same as a discrete uniform distribution but with a continuous random variable
Also a rectangular probability distribution
An even slope upwards as a cumulative distribution function

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8
Q

What is a Bernoulli random variable?

A

One based on the outcome of a trial which produces one of two outcomes (binomial outcomes), interpreted as 1 or 0
p(1) = p
p(0) = 1 - p

In n trials, we can have 0 to n successes
If each trial is a random variable, then the number of successes in n trials is also a random variable, known as a binomial random variable

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9
Q

What is a binomial random variable?

A

The number of successes in n Bernoulli trials
Assumption:
1. p is constant for all trials
2. Trials are independent

A binomial random variable has a distribution completely described by 2 parameters
x ~ B(n, p)

  • To find how many successes (x ) are in n trials we can use nCr
    Because the order doesn’t matter

-[ When we ask how probable is it to have x successes in n trials we can do:
p^x (1 - p)^(n - x)

  • We multiply nCr by this to get the probability distribution function for a binomial random variable

n! / ((n - x)! x!) * p^x (1 - p)^(n - x)

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10
Q

Why when we’re calculating probability are we only interested in the tails?

A

If we continue counting up past the mid point of the probability distribution we would misinterpret it
Such that we would deduce that achieving the top figure has a 100% chance
We have to count in the direction from the centre toward to tail

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11
Q

How do we calculate mean and variance for Bernoulli and binomial distributions?

A

For Bernoulli,
mean = p
variance = p (1 - o)

For Binomial,
mean = np
binomial = np (1 - p)

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12
Q

What is the central limit theorem?

A

The distribution of a large number of independent random variables with finite variance is approximately normal

Let’s say we take a whole bunch of samples of random variables that are not related to each other and find their means
The distribution of these means will be approximately normal

The central limit theorem tells us that because of this result a lot of data tends to be normally distributed

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13
Q

What is a standard normal distribution?

A

A distribution where we have set the mean to 0 and standard deviation to 1
We may want to standardise our values (if they fall into an approximately normal distribution) and turn it into a standard normal distribution to allow data processing (using things like ML) and cross comparison

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14
Q

Why do we use a normal distribution to model asset returns but not asset prices?

A

We use a normal distribution to model continuously compounded asset returns
We do not use it to model asset prices because the left tail of a nd goes to negative infinity, whereas asset prices go to 0

Asset returns are approximately normally distributed, so we can use nd to model (“close enough”)
However asset returns tend to be more kurtotic than normal (longer tails), and options add skew (pos/neg)
There is a lot more of this at L3

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15
Q

What are the characteristics of nd?

A

A normal distribution has these 3 characteristics:
1. Described by 2 parameters, mu and sigma squared (population variance). The formula is X ~ N(mu, sigma squared)

  1. Skew = 0 and kurtosis = 3 (K sub-c = 0). Therefore median = median = mode
  2. A linear combination of 2 or more normal random variables is also normally distributed.
    So R sub-p = w sub-1 R sub-1 + w sub-2 R sub-2 + w sub-3 R sub-3 …. is also nd, althought it is multivariate. Each of these terms is a univariate random variable
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16
Q

What 3 lists of variables define a multivariate normal distribution in a portfolio management context?

A
  1. All the mean returns of all the individual securities (n returns)
  2. All the securities’ variances (n variances)
  3. All pairwise correlations. There are (n^2 - n)/2 unique correlations

Usually in PM we do this at the asset class level, because if we did this at the level of individual securities it could quickly become unmanageable

17
Q

For a nd how many sd are required to capture 95% and 99% of outcomes?

A

95% = 1.96 standard deviations for a normal distribution
99% = 2.58 standard deviations for a normal distribution

18
Q

How do we standardise a normal distribution?

A

We need to set mean to 0 and standard deviation to 1

Let’s say we have a distribution of n=30
Our mean is 4.7
Our standard deviation is 3

For each obseration, we calculate z (the standardised value) as:
z = x sub-i - x-bar / sigma

z = 7.2 - 4.7 / 3 = 0.8333

19
Q

How do we calculate probabilites on a normal distribution?

A

If we have a z-value and excel it’s easy
We can use the function NORM.S.DIST(z, 1)

Where z is the z-value
And 1 means that we use a cumulative probability function rather than a probability density function

The output will be the probability from 0 to 1

If we want a z-value out, we can use NORM.S.INV(probability). This will output a z value from 0 to infinity, but most z values fall between 0 and 3. z-values can also be negative, but because the normal distribution is symmetrical we don’t need to worry about this

20
Q

How are NORM.S.INV(0.95) and NORM.S.INV(0.05) related?

A

NORM.S.INV(0.95) willl return the z-value for 95th percentile
1 - NORM.S.INV(0.95) = 5th percentile

21
Q

How do we find the probability that the return on a portfolio is greater than or equal to 12%, but less than or equal to 20%?

Mean return is 12%
SD is 22%

A

We can express this as:
P(12% =< R sub-p =< 20%)

We calculate the z-value as:
z = (x sub-i - x-bar) / sd

so z = ((20-12)/22) = 0.3636
and z = ((12-12)/22) = 0

Then to find the probabilities we use the NORM.S.DIST function and subtract one from the other:

NORM.S.DIST(0.3636, 1) - NORM.S.DIST(0,1)

22
Q

Why might we use a t-test over a z-test?

A

Student’s t distribution has fatter tails than the normal distribution (excess kurtosis / platykurtic)
Therefore if something is significant in a t-test it will definitely be significant in a z-test

23
Q

What are degrees of freedom?

A

Sample size minus 1 (or n - 1)

As degrees of freedom increases the tails of the t distribution are pulled in and added to the head, such that it converges to a normal distribution over n=200
Thus theoretically we would use a t test for small n values (below 200) and a z-test or normal distribution test for values above 200
However in practice we just use t really

24
Q

What are the test statistics for z-test and t-test?

A

z = (x-bar - mu) / (sigma / standard error)
where mu and sigma are population parameters. As such, only 1 estimate is used

t = (x-bar - mu) / (sigma / standard error)
Where x-bar and s are sample statistics. As such, 2 estimates are used

T-tests are used for hypothesis testing since they are more conservative, more stringent, and produce wide confidence intervals

25
Q

What is the chi-squared distribution?

A

A distribution of variance
The interesting thing about variance is you can’t have a negative value, because it’s deviations squared.
Like log normal, it is bounded below by 0.
Variance follows a very particular distribution, depending of number of parameters used to arrive at the distribution.
The distribution of variances is: the sum of the squares (of deviations) of k independent standard normally distributed random variables.

Degrees of freedom is n - 1, same as t-distribution
Because variance cannot be negative, the distribution flattens out. And with low degres of freedom (2, 3) the distribution gets pushed up against the y-axis
As such, as degrees of freedom increase, the distribution becomes more symmetrical and bell-shaped (though flattening)

26
Q

What is the F-distribution?

A

Bounded below by 0 like the chi square distribution
Because it is the ratio of 2 chi square variables

F = ((chi square sub1)/n sub1 - 1) / ((chi square sub2) / n sub2 - 1)

By convention, the larger figure is used as the numerator on top

F test is used in regression to test the significant of the whole regression. It is explained variance divided by unexplained variance. The higher the number is, the better the model is explaining all total variance.

27
Q

What are the excel functions for chi square distribution?

A

CHISQ.DIST(chi squared value, degrees of freedom)
Input is chi square value, output is a probability

CHISQ.INV(p, degrees of freedom)
Input is a probability, output is a chi square value

28
Q

What are the excel functions for cumulative t distribution?

A

T.DIST(t-value, degrees of freedom, 1)
Use 1 to specify cumulative distribution function.
Input is a t-value, output is a probability

T.INV(p, degrees of freedom)
Input is a probability, output is a t-value

29
Q

What are the excel functions for f distribution?

A

F.DIST(F-value, df for numerator, df for denominator, 1)

Input an f value and the degrees of freedom of the two variables, output is a probability

F.INV(p, df1, df2)

Input a probability and degrees of freedom for the variables, output is an f-value