5.5 Chapter Summary

Interactive Tools:¶

1. Tail Visualizer - Use this tool to visualize tails on log scales.

Tails¶

Definitions and examples are all available in Section 5.1.

1. The tail of a distribution is the part of a distribution associated with unusually extreme events.

2. The survival function is the function $\text{Pr}(X > x)$ as a function of $x$.

3. We derived the survival function and CDF of a geometric random variable.

4. We derived the value of a geometric series: $\sum_{n = 0}^{\infty} r^n = (1 - r)^{-1}$ if $|r| < 1$.

5. We showed that the harmonic series: $\sum_{n = 1}^{\infty}n^{-1}$ diverges.

Asymptotic Rates¶

Definitions and examples are all available in Section 5.2.

1. A sequence is an infinitely long, order list, of numbers.

2. We can visualize convegergence rates using log-log plots, which convert power laws, $a(n) = c n^{-\gamma}$ into lines whose slope is the exponent, $-\gamma$.

3. We can compare how quickly series converge to zero, or diverge to infinity, by studying the limiting ratio:

   $$L = \lim_{n \rightarrow \infty} \frac{a(n)}{b(n)}$$

   (1)

   • If $L = 0$ then $a(n)$ converges to zero faster than $b(n)$

   • If $0 < L < \infty$ then $a(n)$ converges at the same rate as $b(n)$

   • If $L = \infty$ then $a(n)$ slower than $b(n)$

   • If $a(n)$ is faster than, or as fast as, $b(n)$ then $a(n) = \mathcal{O}(b(n))$ (is on the order of $b(n)$)

4. Sometimes it is helpful to evaluate the limit using L’Hopital’s Rule (e.g. use the limiting ratio of the derivatives)

5. Exponential sequences, $r^n$ for $|r| < 1$ converge faster than all power laws $n^{-\gamma}$ for $\gamma > 0$.

6. A distribution whose tails decay slowly (e.g. power laws) is heavy tailed.

Convergence Tests for Series¶

Definitions and examples are all available in Section 5.3.

We considered three tests:

1. The Integral Comparison Test

   • Compare the series to a related integral. The series converges if and only if the integral converges.

2. The Direct and/or Limit Comparison Test

   • Compare the sequences $a$ and $b$.

   • If $a$ converges to zero faster than, or as fast as, $b$, and the series defined by $b$ converges, then so does the series defined by $a$.

   • If $a$ converges to zero slower than, or as slowly as, $b$, and the series defined by $b$ diverges, then so does the series defined by $a$.

3. The Ratio Test

   • Evaluate the limiting ratio: $r = \lim_{n \rightarrow \infty} \frac{|a(n+1)|}{|a(n)|}$ to compare to a geometric series.

Sorting Distributions by their Tails¶

Definitions and examples are all available in Section 5.3.

1. Distributions whose tails are slower than exponential (superexponential) have heavy tails. Examples are power laws, Pareto distributions, and Student’s t distributions.

2. Distributions whose tails decay at an exponential rate include the geometric and exponential distributions.

3. Distributions whose tails decay faster than exponential (subexponential) have light tails. Examples are the Poisson and Normal distributions.