4.2 Expectation and Variance

The mathematical expectation or the mean $\mathop{\text{\rm\sf E}}[X]$ of a real random variable $X$ is a measure for the location of the distribution of $X$. Adding to $X$ a real constant $c$, it holds for the expectation: $\mathop{\text{\rm\sf E}}[X+c] = \mathop{\text{\rm\sf E}}[X] + c$, i.e. the location of the distribution is translated. If $X$ has a density $p(x)$, its expectation is defined as:

$$\mathop{\text{\rm\sf E}}(X) = \int^{\infty}_{- \infty} \, xp(x) dx .$$

If the integral does not exist, neither does the expectation. In practice, this is rather rarely the case.

Let $X_1,\ldots,X_n$ be a sample of identically independently distributed (i.i.d.) random variables (see Section 3.4) having the same distribution as $X$, then $\mathop{\text{\rm\sf E}}[X]$ can be estimated by means of the sample mean:

$$\hat{\mu} = \frac{1}{n} \sum_{t=1}^n X_t.$$

A measure for the dispersion of a random variable $X$ around its mean is given by the variance $\mathop{\text{\rm Var}}(X)$:

$$\mathop{\text{\rm Var}}(X) = \mathop{\text{\rm\sf E}}[(X - \mathop{\text{\rm\sf E}}X)^2]$$

$$=$$ Variance mean squarred deviation of a random variable

$$\quad$$ around its expectation.

If $X$ has a density $p(x)$, its variance can be computed as follows:

$$\mathop{\text{\rm Var}}(X) = \int^{\infty}_{- \infty} \, (x - \mathop{\text{\rm\sf E}}X)^2 p(x) dx .$$

The integral can be infinite. There are empirical studies giving rise to doubt that some random variables appearing in financial and actuarial mathematics and which model losses in highly risky businesses dispose of a finite variance.

As a quadratic quantity the variance's unity is different from that of $X$ itself. It is better to use the standard deviation of $X$ which is measured in the same unity as $X$:

$$\sigma (X) = \sqrt{\mathop{\text{\rm Var}}(X)} \, .$$

Given a sample of i.i.d. variables $X_1,\ldots,X_n$ which have the same distribution as $X$, the sample variance can be estimated by:

$$\hat{\sigma}^2=\frac{1}{n} \sum_{t=1}^n (X_t-\hat{\mu})^2.$$

A $N (\mu, \sigma^2)$ distributed random variable $X$ has mean $\mu$ and variance $\sigma^2.$ The $2\sigma$ area around $\mu$ contains with probability of a little more than 95% observations of $X$:

$$\P(\mu - 2 \sigma < X < \mu + 2 \sigma ) \approx 0.95 \, .$$

A lognormally distributed random variable $X$ with parameters $\mu$ and $\sigma^2$ has mean and variance

$$\mathop{\text{\rm\sf E}}(X) = e^{\mu + \frac{1}{2} \sigma^2} , \quad \mathop{\text{\rm Var}}(X) = e^{2 \mu + \sigma^2} (e^{\sigma^2} - 1) .$$

A $B (n,p)$ distributed variable $X$ has mean $np$ and variance $np(1-p)$. The approximation (3.1) is chosen such that the binomial and normal distribution have identical mean and variance.