# 3.3 Parameters

Parameters describe random vectors much as we might use height or age to describe a person. Formally, a **parameter** is a function that is applied to a random vector’s probability distribution. It may take on real, vector, or matrix values. A standard deviation, mean vector, or covariance matrix are all examples of parameters. In this section, we describe parameters for random variables. In Section 3.4, we extend the discussion to parameters for random vectors.

###### 3.3.1 Expectation

Let *X* be a random variable. We denote the** expected value**, **expectation**, or **mean** of *X* as either μ or *E*(*X*). If *X* is discrete, we define its expectation as

[3.3]

where ϕ is the PF of *X*. If *X* is continuous, we replace the summation with an integral and define

[3.4]

where ϕ is the PDF of *X*.

Expectation is used to define a number of other parameters, but first we must discuss expectations of functions of random variables.

###### 3.3.2 Expectation of a function of a random variable

Suppose *X* is a random variable and *f* is a function from to . Then *f *(*X*) is a new random variable2 whose probability distribution we can, at least in theory, infer from that of *X*. We do not need the probability distribution of *f *(*X*) in order to determine the expectation *E*[*f *(*X*)]. This can be obtained directly from the probability distribution of *X* using the formula

[3.5]

or

[3.6]

depending upon whether *X* is discrete or continuous. In a sense, [3.5] and [3.6] are generalizations of [3.3] and [3.4].

###### 3.3.3 Variance and standard deviation

Variance is a parameter that measures how dispersed a random variable’s probability distribution is. In Exhibit 3.1, two PDFs have a mean of 0. The one on the left is more dispersed than the one on the right. It has a higher variance.

The **variance**, denoted α^{2} or *var*(*X*), of a random variable *X* is defined as an expectation of a function of *X*:

[3.7]

S**tandard deviation**, denoted σ or *std*(*X*), is the positive square-root of variance.

###### 3.3.4 Skewness

**Skew** or **skewness** is a measure of asymmetry in a random variable’s probability distribution. Both PDFs in Exhibit 3.2 have the same mean and standard deviation. The one on the left is positively skewed. The one on the right is negatively skewed.

The skewness of a random variable *X* is denoted η_{1} or *skew*(*X*). It is defined as

[3.8]

###### 3.3.5 Kurtosis

**Kurtosis** is another parameter that describes the shape of a random variable’s probability distribution. Consider the two PDFs in Exhibit 3.3. Both have a mean and skewness of 0. Which would you say has the greater standard deviation? It is impossible to say. The distribution on the right is more peaked at the center, which might lead us to believe that it has a lower standard deviation. It has fatter tails, which might lead us to believe that it has a greater standard deviation. If the effect of the peakedness exactly offsets that of the fat tails, the two distributions may have the same standard deviation. The different shapes of the two distributions illustrates kurtosis. The distribution on the right has a greater kurtosis than the distribution on the left.

The **kurtosis** of a random variable *X* is denoted η_{2} or *kurt*(*X*).3 It is defined as

[3.9]

If a distribution’s kurtosis is greater than 3, it is said to be **leptokurtic**. If its kurtosis is less than 3, it is said to be** platykurtic**. Leptokurtosis is associated with distributions that are simultaneously “peaked” and have “fat tails.” Platykurtosis is associated with distributions that are simultaneously less peaked and have thinner tails. In Exhibit 3.3, the distribution on the left is platykurtic. The one on the right is leptokurtic.

###### 3.3.6 Quantiles

Consider a random variable *X* with CDF Φ. A *q*-quantile of *X* is any value *x* such that *Pr*(*X* ≤ *x*)* = q*. A *q*-quantile need not exist. If it does exist, it need not be unique.4 In most VaR applications, all *q*-quantiles exist and are unique for *q* ∈ (0,1). In such cases, a *q*-quantile is a parameter and equals the inverse CDF evaluated at *q*. For this reason, we denote a *q*-quantile as Φ^{–1}(*q*).

The term “percentile” may be used in place of “quantile”. A percentile is a quantile expressed as a percentage. For example, a 95^{th} percentile is a .95 quantile.

###### 3.3.7 Moments

For any positive integer *k*, the *k*^{th} **moment** of a random variable *X* is defined as

[3.10]

Its *k*^{th} **central moment** is defined as

[3.11]

where μ = *E*(*X*). Based upon our earlier definitions, the expectation and variance of a random variable are its first moment and second central moment. Its skewness and kurtosis are scaled third and fourth central moments.

For any *n* > 0, a random variable’s first *n* moments convey the same information as its first *n* central moments—each can be derived from the other. See Exercise 3.15.

We say a random variable *X* is **bounded** if there exists a number *a* such that *Pr*(|*X*| > *a*) = 0. If a random variable is bounded, all its moments exist. If it is unbounded, specific moments may or may not exist. However, if the *k*^{th} moment of *X* exists, then all moments of order less than *k* must also exist.

###### Exercises

PDFs for two continuous random variables are illustrated in Exhibit 3.4. Assume probability density is 0 for both distributions outside the graphed regions. Where possible, indicate which random variable has the greater:

- expectation,
- standard deviation,
- skewness,
- kurtosis, and
- .25-quantile.

Consider a discrete random variable *Y*, which represents the number of “heads” that will be obtained in three flips of a fair coin. It has PF

[3.12]

- Calculate the mean of
*Y*. - Calculate the variance of
*Y*. - Calculate the standard deviation of
*Y*. - Calculate the skewness of
*Y*. - Calculate the kurtosis of
*Y*. - Calculate a .10 quantile of
*Y*. - Calculate a .875 quantile of
*Y*.

Consider a continuous random variable *Z* with PDF

[3.13]

- Calculate the mean of
*Z*. - Calculate the variance of
*Z*. - Calculate the standard deviation of
*Z*. - Calculate the skewness of
*Z*. - Calculate the kurtosis of
*Z*. - Calculate a .10-quantile of
*Z*. - Calculate a .875-quantile of
*Z*.

Consider the random variable *W* = *Z* ^{2}, where Z is defined as in the previous exercise.

- Calculate the mean of
*W*. - Calculate the variance of
*W*. - Calculate the standard deviation of
*W*.

True or false: If a continuous random variable *X* has a symmetric distribution, ϕ(*x* – μ) = ϕ(–*x* – μ), it must have 0 skewness.

Solution

In general, for any random variable *X* and any constant *b*, *E*(*bX*) = *bE*(*X*). Prove this result for the case *X* is discrete.

Solution

In general, for any random variable *X* and any constant *a*, *E*(*X* + *a*) = *E*(*X*) + *a*. Prove this result for the case *X* is continuous.

Solution

In general, for any random variable *X* and any constant *b*, *std*(*bX*) = |*b|std*(*X*), where |*b|* indicates the absolute value of *b*. Prove this result for the case *X* is discrete. Use your result from Exercise 3.6.

Solution

In general, for any random variable *X* and any constant *a*, *std*(*X* + *a*) = *std*(*X*). Prove this result for the case *X* is continuous. Use your result from Exercise 3.7.

Solution