TAOCP 3.3.1: General Test Procedures for Studying Random Data
Section 3.3.1 exercises: 24/25 solved.
Section 3.3.1. General Test Procedures for Studying Random Data
Exercises from TAOCP Volume 2 Section 3.3.1: 24/25 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [00] | immediate | verified | 1m03s |
| 2 | [20] | medium | verified | 1m34s |
| 3 | ▶ [23] | medium | verified | 7m08s |
| 4 | ▶ [23] | medium | verified | 5m53s |
| 5 | [22] | medium | verified | 10m02s |
| 6 | [M20] | math-medium | verified | 2m41s |
| 7 | [M16] | math-medium | solved | 6m48s |
| 8 | [00] | immediate | verified | 2m22s |
| 9 | ▶ [**] | verified | 4m25s | |
| 10 | [**] | verified | 3m15s | |
| 11 | [**] | verified | 10m50s | |
| 12 | [**] | verified | 12m47s | |
| 13 | [**] | verified | 6m46s | |
| 14 | ▶ [**] | verified | 3m11s | |
| 15 | [**] | verified | 7m32s | |
| 16 | ▶ [**] | verified | 1m52s | |
| 17 | [**] | solved | 8m31s | |
| 18 | [**] | verified | 5m16s | |
| 19 | [**] | verified | 6m18s | |
| 20 | [**] | verified | 9m46s | |
| 21 | [**] | verified | 2m28s | |
| 22 | [**] | verified | 6m44s | |
| 23 | [**] | verified | 1m53s | |
| 24 | ▶ [**] | verified | 2m29s | |
| 25 | [**] | - | - |
TAOCP 3.3.1 Exercise 1
Equation (5) arises from the dice-throwing experiment with eleven categories, namely the possible sums $2,3,\ldots,12$.
TAOCP 3.3.1 Exercise 2
Let the two dice be labeled die A and die B.
TAOCP 3.3.1 Exercise 3
To test whether the dice are fair, we must use the probability distribution for the sum of two ordinary dice.
TAOCP 3.3.1 Exercise 4
Let the first die be fair, with outcomes $1,2,3,4,5,6$ equally likely, each with probability $\frac{1}{6}$.
TAOCP 3.3.1 Exercise 5
Let the observations be arranged in increasing order: \begin{aligned} X_{(1)}&=0.
TAOCP 3.3.1 Exercise 6
Let $F_n(x)$ be the empirical distribution function defined by equation (10) of Section 3.
TAOCP 3.3.1 Exercise 7
Let $F_n(x)$ be the empirical distribution function based on $n$ independent observations $X_1, \dots, X_n$ from a **continuous** distribution $F(x)$.
TAOCP 3.3.1 Exercise 8
Each of the 20 values of $K_{10}^+$ was itself computed from 10 observations.
TAOCP 3.3.1 Exercise 9
**Solution (corrected)** We are asked to discuss the merits of pooling the 20 values of $K_{10}^{+}$ with the 20 values of $K_{10}^{-}$ and then applying a Kolmogorov-Smirnov test to the resulting 40...
TAOCP 3.3.1 Exercise 10
Let the original observations produce counts $Y_1,\ldots,Y_k$, with probabilities $p_1,\ldots,p_k$.
TAOCP 3.3.1 Exercise 11
Let F_n(x)=\frac{1}{n}\#\{j:X_j\le x\} be the empirical distribution function of the original sample
TAOCP 3.3.1 Exercise 12
**Exercise 3.
TAOCP 3.3.1 Exercise 13
Let F_n(x)=\frac1n\#\{x_j\le x\}, and let
TAOCP 3.3.1 Exercise 14
Let $Y_s=np_s+\sqrt n\,Z_s,\qquad s=1,\ldots,k,$ where the variables $Z_s$ are bounded as $n\to\infty$.
TAOCP 3.3.1 Exercise 15
Let J_n=\det\!
TAOCP 3.3.1 Exercise 16
We write $\gamma(x+1, x + z\sqrt{2x} + p) = \int_0^{x + z\sqrt{2x} + p} e^{-t} t^x \, dt / x!.$ Setting $t = x + s\sqrt{2x}$, we have $dt = \sqrt{2x}, ds$, and the integral becomes $\frac{\sqrt{2x}}{x...
TAOCP 3.3.1 Exercise 17
**Exercise 3.
TAOCP 3.3.1 Exercise 18
The statistics $K_n^+$ and $K_n^-$ are defined as the maximum positive and maximum negative deviations, respectively, of the empirical distribution function from the theoretical distribution function.
TAOCP 3.3.1 Exercise 19
Let X^{(1)},X^{(2)},\ldots,X^{(n)} be independent observations from an unknown distribution on $\mathbf R^s$, where
TAOCP 3.3.1 Exercise 20
**Exercise 3.
TAOCP 3.3.1 Exercise 21
Suppose that $F(x)$ has jumps at points $\xi_1,\xi_2,\ldots$, with jump sizes $\Delta_j=F(\xi_j)-F(\xi_j-0)>0.$ If the sample contains $m_j$ occurrences of $\xi_j$, then the empirical distribution fun...
TAOCP 3.3.1 Exercise 22
Let D_n(u)=\sqrt n\,(G_n(u)-u), \qquad 0\le u\le1, where $G_n$ is the empirical distribution function of the transformed variables
TAOCP 3.3.1 Exercise 23
Let $U_i=F(X_i)$, and let the values be renumbered so that U_1\le U_2\le\cdots\le U_n.
TAOCP 3.3.1 Exercise 24
For fixed $(p,q,r)$ with $p+q+r=1$, the exact distribution of $V$ is obtained by enumerating all triples $(Y_1,Y_2,Y_3)$ satisfying $Y_1+Y_2+Y_3=n$.