diff --git a/MSMLLN/MSMLLN.py b/MSMLLN/MSMLLN.py
new file mode 100644
index 0000000..b5d2054
--- /dev/null
+++ b/MSMLLN/MSMLLN.py
@@ -0,0 +1,22 @@
+# imports
+import numpy as np
+import matplotlib.pyplot as plt
+
+# plotting settings
+plt.style.use('seaborn')
+plt.rcParams['figure.figsize'] = (16, 8)
+
+np.random.seed(0)
+mu, sigma = 0, 1 # mean, standard deviation
+x = np.random.normal(mu, sigma, 1000)
+cum_sum = np.cumsum(x)
+sample_size = np.arange(1, len(x)+1)
+averges = np.divide(cum_sum, sample_size)
+
+plt.xlabel('Size of a sample', fontsize=16)
+plt.ylabel('Average', fontsize=16)
+plt.axhline(0, ls= '--', c='black', label='mean')
+plt.plot(sample_size, averges,
+ c='r', lw=5, alpha=0.6, label='sample average')
+plt.legend(loc="upper right", frameon=True, prop={'size': 14})
+plt.title('Law of Large Numbers', fontsize=20);
diff --git a/MSMLLN/MSMLLN.png b/MSMLLN/MSMLLN_R.png
similarity index 100%
rename from MSMLLN/MSMLLN.png
rename to MSMLLN/MSMLLN_R.png
diff --git a/MSMLLN/MSMLLN_py.png b/MSMLLN/MSMLLN_py.png
new file mode 100644
index 0000000..b608af8
Binary files /dev/null and b/MSMLLN/MSMLLN_py.png differ
diff --git a/MSMLLN/Metainfo.txt b/MSMLLN/Metainfo.txt
index aa35b58..e244598 100755
--- a/MSMLLN/Metainfo.txt
+++ b/MSMLLN/Metainfo.txt
@@ -3,9 +3,7 @@ Name of QuantLet : MSMLLN
Published in : MSM
-Description : 'Plots the points showing law of large numbers.
-As the sample size becomes larger, the sample mean converges to
-the theoretical (true) mean of the distribution.'
+Description : 'Plots the points showing law of large numbers. As the sample size becomes larger, the sample mean converges to the theoretical (true) mean of the distribution.'
Keywords :
- pdf
@@ -21,8 +19,8 @@ See also :
- MSMpdfasymmetric
- MSMasprob
-Author : Xiu Xu
+Author : Xiu Xu, Jacek Wiland
-Submitted : Thu, November 5 2015 by Chen Huang
+Submitted : December 15 2020
Example : Plot of points presenting law of large numbers.
diff --git a/MSMLLN/README.md b/MSMLLN/README.md
index 991f232..8595782 100644
--- a/MSMLLN/README.md
+++ b/MSMLLN/README.md
@@ -1,51 +1,42 @@
+
-[
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"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from scipy.stats import lognorm\n",
+ "\n",
+ "# Parameters for lognormal distribution\n",
+ "mu = 0.5\n",
+ "sigma = 0.3\n",
+ "s = sigma\n",
+ "scale = np.exp(mu)\n",
+ "\n",
+ "# Define the PDF and CDF\n",
+ "def dlognormal_loss(x):\n",
+ " return lognorm.pdf(x, s, scale=scale)\n",
+ "\n",
+ "def plognormal_loss(x):\n",
+ " return lognorm.cdf(x, s, scale=scale)\n",
+ "\n",
+ "# Plot the CDF\n",
+ "x_vals = np.linspace(0, 4, 400)\n",
+ "plt.plot(x_vals, plognormal_loss(x_vals), color = 'blue', label='CDF', linewidth=2)\n",
+ "\n",
+ "# Plot the PDF\n",
+ "plt.plot(x_vals, dlognormal_loss(x_vals), label='PDF', color='black', linestyle='--', linewidth=2)\n",
+ "\n",
+ "# Fill area beyond VaR (shadow region)\n",
+ "var_alpha = 2.42\n",
+ "x_fill = np.linspace(var_alpha, 4, 100)\n",
+ "y_fill = dlognormal_loss(x_fill)\n",
+ "plt.fill_between(x_fill, y_fill, color='grey', alpha=0.5)\n",
+ "\n",
+ "# Add arrow and text for P(S > VaR[alpha]) == alpha\n",
+ "plt.text(3.1, 0.2, r'$P(S > VaR_{\\alpha}) = \\alpha$', fontsize=12)\n",
+ "plt.annotate('', xy=(2.7, 0.05), xytext=(3, 0.2),\n",
+ " arrowprops=dict(arrowstyle='->', lw=3))\n",
+ "\n",
+ "# Mark VaR[alpha]\n",
+ "plt.axvline(x=var_alpha, ymin=0, ymax=0.9, color='black', linestyle='-')\n",
+ "plt.axhline(y=0.9, xmin=0, xmax=var_alpha/4, color='black', linestyle='-')\n",
+ "plt.text(var_alpha, -0.02, r'$VaR_{\\alpha}$', ha='center', va='top')\n",
+ "\n",
+ "# Fill for expected shortfall (striped)\n",
+ "x_es = np.linspace(var_alpha, 4, 100)\n",
+ "y_es = plognormal_loss(x_es)\n",
+ "x_poly = np.concatenate(([0], x_es, [0]))\n",
+ "y_poly = np.concatenate(([0.9], y_es, [plognormal_loss(4)]))\n",
+ "plt.fill(x_poly, y_poly, facecolor='grey', edgecolor='black', hatch='/', linewidth=1, alpha=0.6)\n",
+ "\n",
+ "plt.text(0.3, 0.93, r'$1 - \\alpha = 0.9$', fontsize=12)\n",
+ "plt.text(1.6, 0.95, r'$\\alpha \\cdot ES_{\\alpha}(X)$', fontsize=12)\n",
+ "\n",
+ "# Replot curves for clarity\n",
+ "plt.plot(x_vals, plognormal_loss(x_vals), color='blue', linewidth=2)\n",
+ "plt.plot(x_vals, dlognormal_loss(x_vals), color='black', linestyle='--', linewidth=2)\n",
+ "\n",
+ "# Legend\n",
+ "\n",
+ "\n",
+ "# Labels and display\n",
+ "plt.xlabel('Loss')\n",
+ "plt.ylim(-0.05, 1.05)\n",
+ "plt.xlim(0, 4)\n",
+ "plt.grid(True)\n",
+ "plt.title('Lognormal Loss Distribution: PDF and CDF')\n",
+ "plt.legend(loc='upper left')\n",
+ "plt.show()\n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python [conda env:base] *",
+ "language": "python",
+ "name": "conda-base-py"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.12.7"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/MSMasprob/MSMasprob.py b/MSMasprob/MSMasprob.py
new file mode 100644
index 0000000..1dad933
--- /dev/null
+++ b/MSMasprob/MSMasprob.py
@@ -0,0 +1,27 @@
+# imports
+import numpy as np
+import matplotlib.pyplot as plt
+
+# plotting settings
+plt.style.use('seaborn')
+plt.rcParams['figure.figsize'] = (16, 8)
+
+# parameters
+n = 1000
+k = np.floor(np.log(n)/np.log(2)) - 1
+x = np.zeros([n, 10])
+x[0] = 1
+z = np.array([0.5, 0.2, 0.4, 0.6, 0.8])
+
+for s in range(2, len(z)+1):
+ for i in range(1, int(k+1)):
+ for j in range(0, 2**i + 1):
+ t = 2**i + j
+ x[t, 1] = (j * 2**(-i) <= 0.5) & (0.5 < (j + 1) * 2**(-i))
+ x[t, s] = (j * 2**(-i) <= z[s-1]) & (z[s-1] < (j + 1) * 2**(-i))
+ x = x * 1
+
+# Plot the time series
+plt.plot(x[1:400, 1], 'o', c = 'b', ms=5)
+plt.plot(x[1:400, 2], 'o', c = 'g', ms=5)
+plt.plot(x[1:400, 5], 'o', c = 'r', ms=5);
diff --git a/MSMasprob/MSMasprob.png b/MSMasprob/MSMasprob_R.png
similarity index 100%
rename from MSMasprob/MSMasprob.png
rename to MSMasprob/MSMasprob_R.png
diff --git a/MSMasprob/MSMasprob_py.png b/MSMasprob/MSMasprob_py.png
new file mode 100644
index 0000000..7cc6ac0
Binary files /dev/null and b/MSMasprob/MSMasprob_py.png differ
diff --git a/MSMasprob/Metainfo.txt b/MSMasprob/Metainfo.txt
index 7140696..798734c 100755
--- a/MSMasprob/Metainfo.txt
+++ b/MSMasprob/Metainfo.txt
@@ -3,10 +3,7 @@ Name of QuantLet : MSMasprob
Published in : MSM
-Description : 'Displays an example of which meets the convergence in
-mean square but not convergence in almost sure series of the example.
-We select n = 400 and plot three cases: the initial value of
-Z = 0.5 (blue), 0.2 (green), and 0.8 (dark red).'
+Description : 'Displays an example of which meets the convergence in mean square but not convergence in almost sure series of the example. We select n = 400 and plot three cases: the initial value of Z = 0.5 (blue), 0.2 (green), and 0.8 (dark red).'
Keywords :
- random-number-generation
@@ -21,9 +18,8 @@ See also :
- MSMpdfasymmetric
- SFSmsr1
-Author : Xiu Xu
+Author: Xiu Xu, Jacek Wiland
-Submitted : Thu, November 5 2015 by Chen Huang
+Submitted : December 15 2020
-Example : 'Plot of the series of a random variable satisfying convergence
-in mean square but not convergence in almost sure.'
+Example : 'Plot of the series of a random variable satisfying convergence in mean square but not convergence in almost sure.'
diff --git a/MSMasprob/README.md b/MSMasprob/README.md
index 0ed993b..f7d88b0 100644
--- a/MSMasprob/README.md
+++ b/MSMasprob/README.md
@@ -1,62 +1,41 @@
+
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+
](http://quantlet.de/)
-
-## [
](http://quantlet.de/) **MSMLLN** [
](http://quantlet.de/)
-
-```yaml
-
-Name of QuantLet : MSMLLN
+```
+Name of QuantLet: MSMLLN
-Published in : MSM
+Published in: MSM
-Description : 'Plots the points showing law of large numbers. As the sample size becomes larger,
-the sample mean converges to the theoretical (true) mean of the distribution.'
+Description: Plots the points showing law of large numbers. As the sample size becomes larger, the sample mean converges to the theoretical (true) mean of the distribution.
-Keywords : pdf, normal, distribution, mean, plot, parameter, visualization, discrete
+Keywords:
+- pdf
+- normal
+- distribution
+- mean
+- plot
+- parameter
+- visualization
+- discrete
-See also : MSMpdfasymmetric, MSMasprob
+See also:
+- MSMpdfasymmetric
+- MSMasprob
-Author : Xiu Xu
+Author: Xiu Xu, Jacek Wiland
-Submitted : Thu, November 5 2015 by Chen Huang
+Submitted: December 15 2020
-Example : Plot of points presenting law of large numbers.
+Example: Plot of points presenting law of large numbers.
```
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