====== Week5 - Descriptive Statistics ======

===== 5.1 - Covariance Stationarity =====

  * Recall:
    * \(E[X_t] = \mu\) indep of t.
    * \(var(X_t) = \sigma^2\) indep of t.
    * \(cov(X_t,X_{t-j}) = \gamma_j\) indep of t.
    * \(cor(X_t,X_{t-j}) = \rho_j\) indep of t.

  * In SP500, the capitalization of each company is used to compute the weight for that company in the "SP500 portfolio": \(w_i = \frac{cap_i}{\sum\limits_j cap_j}\)
  * The volativity of SP500 is also called **market volatility**.

  * Descriptive statistics = data summaries (we try to compute some features from the data).

===== 5.2 - Histograms =====

  * We can build blockly histograms or smoothed histograms.
  * We can infer the mean, deviation, skewness and kurtosis from the histograms.

===== 5.3 - Sample Statistics =====

==== Percentiles ====

  * Sample quantile is often called **empirical quantile** or **percentile**.

==== Quartiles ====

  * q.25 is the first quartile
  * q.50 is the second quartile (or median)
  * q.75 is the third quartile
  * q.75 - q.25 is the **interquartile range** (IQR)

==== Historical value-at-risk ====

  * eg. value at risk computed from the data (eg. from empirical quantiles).

==== Sample Statistics ====

  * We can compute Sample mean, variance, std deviation, skewness, kurtosis and excess kurtosis.