Update solutions

11111792 · Marc Sommerhalder · 74223413 · 11111792
Commit 11111792 authored 2 years ago by Marc Sommerhalder
--- a/Musterloesungen/08_Diskrete_Verteilungen_Musterloesungen_R.ipynb
+++ b/Musterloesungen/08_Diskrete_Verteilungen_Musterloesungen_R.ipynb
@@ -562,13 +562,38 @@
    "sum(dist[1 <= dist$k & dist$k <= 3,]$p)"
   ]
  },
+  {
+   "cell_type": "markdown",
+   "id": "c4d34af2",
+   "metadata": {},
+   "source": [
+    "## Übung 10. Empirische Verteilung"
+   ]
+  },
  {
   "cell_type": "code",
-   "execution_count": null,
-   "id": "bd0f4343",
+   "execution_count": 5,
+   "id": "98c2f256",
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[1] 5.48\n"
+     ]
+    }
+   ],
+   "source": [
+    "x <- c(\n",
+    "  5, 6, 8, 10, 1, 2, 9, 10, 3, 4, 9, 5, 3, 9, 3, 5, 7,\n",
+    "  6, 1, 1, 9, 8, 9, 6, 9, 4, 5, 8, 2, 4, 2, 5, 10, 2,\n",
+    "  4, 5, 10, 3, 6, 3, 1, 8, 1, 3, 5, 5, 2, 10, 8, 10\n",
+    ")\n",
+    "histogram <- hist(x, breaks=0.5:10.5, plot=FALSE)\n",
+    "E <- sum(histogram$mids  * histogram$density)\n",
+    "print(E)"
+   ]
  }
 ],
 "metadata": {

 %% Cell type:markdown id:06419e14 tags:

 # 8 - Diskrete Verteilungen

 %% Cell type:markdown id:4b055420 tags:

 ## Übung 2. Empirische Verteilung

 %% Cell type:code id:23ede07d tags:

 ``` R
 x <- c(1, 4, 2, 2, 4, 3, 1, 2, 2, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 5)

 histogram <- hist(x, breaks=0.5:5.5, probability=TRUE)
 histogram$counts <- cumsum(histogram$counts)
 histogram$counts <- histogram$counts / max(histogram$counts)
 plot(histogram)
 ```

 %% Output





 %% Cell type:code id:f9cf4327 tags:

 ``` R
 histogram <- hist(x, breaks=0.5:5.5, plot=FALSE)
 E <- sum(histogram$mids  * histogram$density)
 S <- sum(histogram$mids^2 * histogram$density) - E^2
 print(c(E, S))
 ```

 %% Output

    [1] 2.80 1.86

 %% Cell type:markdown id:3e18a91f tags:

 ## Übung 5. Binomialverteilung

 %% Cell type:code id:55ce6ecd tags:

 ``` R
 dbinom(0, 10, 0.05)
 ```

 %% Output

    0.598736939238379
    0.598736939238379

 %% Cell type:code id:15a4c19c tags:

 ``` R
 dbinom(1, 10, 0.05)
 ```

 %% Output

    0.315124704862305
    0.315124704862305

 %% Cell type:code id:4202c5f8 tags:

 ``` R
 dbinom(2, 10, 0.05)
 ```

 %% Output

    0.0746347985200195
    0.0746347985200195

 %% Cell type:code id:0f4191b2 tags:

 ``` R
 1 - pbinom(2, 10, 0.05)
 ```

 %% Output

    0.0115035573792969
    0.0115035573792969

 %% Cell type:markdown id:97c95e85 tags:

 ## Übung 6. Poissonverteilung

 %% Cell type:code id:10a26cd1 tags:

 ``` R
 dpois(3, 5)
 ```

 %% Output

    0.140373895814281
    0.140373895814281

 %% Cell type:code id:93ac0221 tags:

 ``` R
 1 - ppois(9, 5)
 ```

 %% Output

    0.0318280573062049
    0.0318280573062049

 %% Cell type:code id:5018483c tags:

 ``` R
 padd <- function (a, b) {a + b - a * b}
 padd(1 - ppois(9, 5), 1 - ppois(7, 4))
 ```

 %% Output

    0.081334189445324
    0.081334189445324

 %% Cell type:code id:fe96ddab tags:

 ``` R
 1 - ppois(16, 9)
 ```

 %% Output

    0.0111059093775838
    0.0111059093775838

 %% Cell type:markdown id:43a530b0 tags:

 ## Übung 7. Hypergeometrische Verteilung

 %% Cell type:code id:4f8102b9 tags:

 ``` R
 k <- 0:6
 p <- sapply(k, function (x) dhyper(x, 6, 42 - 6, 6))
 dist <- data.frame(k, p)
 dist
 ```

 %% Output

    
    A data.frame: 7 × 2
    
    | k &lt;int&gt; | p &lt;dbl&gt; |
    |---|---|
    | 0 | 3.713060e-01 |
    | 1 | 4.311941e-01 |
    | 2 | 1.684352e-01 |
    | 3 | 2.722185e-02 |
    | 4 | 1.801446e-03 |
    | 5 | 4.117591e-05 |
    | 6 | 1.906292e-07 |
    
    A data.frame: 7 × 2
    \begin{tabular}{ll}
     k & p\\
     <int> & <dbl>\\
    \hline
    	 0 & 3.713060e-01\\
    	 1 & 4.311941e-01\\
    	 2 & 1.684352e-01\\
    	 3 & 2.722185e-02\\
    	 4 & 1.801446e-03\\
    	 5 & 4.117591e-05\\
    	 6 & 1.906292e-07\\
    \end{tabular}

 %% Cell type:code id:90922910 tags:

 ``` R
 1 - phyper(2, 6, 42 - 6, 6)
 sum(dist[3 <= dist$k & dist$k <= 6,]$p)
 ```

 %% Output

    0.0290646625691555
    0.0290646625691555

    0.0290646625691555
    0.0290646625691555

 %% Cell type:code id:522939ab tags:

 ``` R
 phyper(5, 6, 42 - 6, 6)
 sum(dist[0 <= dist$k & dist$k <= 5,]$p)
 ```

 %% Output

    0.999999809370798
    0.999999809370798

    0.999999809370797
    0.999999809370797

 %% Cell type:code id:3c04e5b7 tags:

 ``` R
 phyper(4, 6, 42 - 6, 6) - phyper(0, 6, 42 - 6, 6)
 sum(dist[1 <= dist$k & dist$k <= 3,]$p)
 ```

 %% Output

    0.628652598485718
    0.628652598485718

    0.626851152525093
    0.626851152525093

-%% Cell type:code id:bd0f4343 tags:
+%% Cell type:markdown id:c4d34af2 tags:
+
+## Übung 10. Empirische Verteilung
+
+%% Cell type:code id:98c2f256 tags:

 ``` R
+x <- c(
+  5, 6, 8, 10, 1, 2, 9, 10, 3, 4, 9, 5, 3, 9, 3, 5, 7,
+  6, 1, 1, 9, 8, 9, 6, 9, 4, 5, 8, 2, 4, 2, 5, 10, 2,
+  4, 5, 10, 3, 6, 3, 1, 8, 1, 3, 5, 5, 2, 10, 8, 10
+)
+histogram <- hist(x, breaks=0.5:10.5, plot=FALSE)
+E <- sum(histogram$mids  * histogram$density)
+print(E)
 ```
+
+%% Output
+
+    [1] 5.48