<jupyter_start><jupyter_text>Lecture 16 Exponential Distribution, Memoryless Property  Exponential Distribution DescriptionReal-valued distribution describing wait times, survival times. Rate parameter $\lambda \gt 0$. Continuous analog of the geometric distribution.Unique in that the exponential distribution has the memoryless property.<jupyter_code>import matplotlib
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline  

plt.xkcd()

x = np.linspace(0,5, 500)

_, ax = plt.subplots()

lambdas = [1.5, 1.0, 0.5]
for l in lambdas:
    y = l * np.e ** (l*-x)
    ax.plot(x, y, label="$\lambda = ${}".format(l))
ax.grid(True, which='both')
ax.axhline(y=0, color='k')
ax.axvline(x=0, color='k')

legend = ax.legend()
for label in legend.get_texts():
    label.set_fontsize('large')
for label in legend.get_lines():
    label.set_linewidth(1.5)

plt.show()<jupyter_output><empty_output>
{'fraction_non_alphanumeric': 0.1319290465631929, 'fraction_numerical': 0.019955654101995565, 'mean_word_length': 5.02, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '1', 'dir': 'jupyter-structured-clean-dedup', 'id': '445902', 'license': 'BSD-3-Clause', 'n_tokens_mistral': 326, 'n_tokens_neox': 311, 'n_words': 87, 'path': 'Lecture_16.ipynb', 'repo_name': 'dirtScrapper/Stats-110-master', 'size': '902'}