# 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.