Verteilungsstatistiken für das 2. Tutorium
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Jacob.Fritsche committed
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "9b437a81-b193-48ae-91bf-48edcfb12e18",
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"metadata": {},
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"outputs": [],
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"source": [
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"#pip install numpy matplotlib scipy"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "a4ea5fe4-eaa9-4cc4-8677-385a68577332",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import scipy.stats as stats"
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]
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},
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{
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"cell_type": "markdown",
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"id": "df08071e-cd3c-49dd-8623-baf38082eab0",
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"metadata": {},
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"source": [
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"Generate a normal distribution"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 52,
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"id": "280f78b3-4338-49ec-8929-732d429762db",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Generate 1000 random numbers from a normal distribution\n",
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"mu, sigma = 3, 6 # mean and standard deviation\n",
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"data = np.random.normal(mu, sigma, 1000)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "7f0ec0a0-ff5f-4fb1-93ae-509cf3d3b066",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Create a histogram of the data\n",
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"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
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"title = \"Histogram with random numbers from: mu = %.2f, std = %.2f\" % (mu, sigma)\n",
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"plt.title(title)\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"id": "0df9c2e5-ae6d-48c0-93a4-12a6e70240c2",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"# Show the approximation with the normal distribution\n",
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"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
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"xmin, xmax = plt.xlim()\n",
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"x = np.linspace(xmin, xmax, 100)\n",
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"p = stats.norm.pdf(x, mu, sigma)\n",
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"plt.plot(x, p, 'k', linewidth=2)\n",
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"title = \"Fit results: mu = %.2f, std = %.2f\" % (mu, sigma)\n",
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"plt.title(title)\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "817835f6-baf8-4b03-8136-426c81b7b4fa",
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"metadata": {},
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"outputs": [],
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"source": [
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"#Show the second normal distribution\n",
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"mu2, sigma2 = 1, 1\n",
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"p2 = stats.norm.pdf(x, mu2, sigma2)\n",
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"plt2.plot(x, p2, 'r', linewidth=2)\n",
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"title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu, sigma, mu2, sigma2)\n",
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"plt2.title(title)\n",
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"plt2.show()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f3c0c504-e041-4d85-8da0-3ef766f8e87a",
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"metadata": {},
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"source": [
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"Generate a second normal distribution"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"id": "f7118230-fc88-4a71-b813-2daa0c63c97c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Generate 1000 random numbers from the first normal distribution\n",
|
||||
"mu1, sigma1 = 3, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"data = np.random.normal(mu1, sigma1, 1000)\n",
|
||||
"\n",
|
||||
"# Create a histogram of the data\n",
|
||||
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
|
||||
"\n",
|
||||
"# Add a graph of the first normal distribution used to generate the data\n",
|
||||
"xmin, xmax = -1, 5\n",
|
||||
"x = np.linspace(xmin, xmax, 100)\n",
|
||||
"p1 = stats.norm.pdf(x, mu1, sigma1)\n",
|
||||
"plt.plot(x, p1, 'k', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
|
||||
"mu2, sigma2 = 1, 0.4\n",
|
||||
"p2 = stats.norm.pdf(x, mu2, sigma2)\n",
|
||||
"plt.plot(x, p2, 'r', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add vertical lines at x = 1 and x = 3 with labels\n",
|
||||
"plt.axvline(x=1, color='b', linestyle='--', label='mu theoretical distribution')\n",
|
||||
"plt.axvline(x=3, color='m', linestyle='--', label='mu in sample')\n",
|
||||
"\n",
|
||||
"title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu1, sigma1, mu2, sigma2)\n",
|
||||
"plt.title(title)\n",
|
||||
"plt.xlim(xmin, xmax)\n",
|
||||
"plt.legend()\n",
|
||||
"\n",
|
||||
"# Show the plot\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 22,
|
||||
"id": "8b4766b1-6fb5-487b-b552-790f43f83707",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Generate 1000 random numbers from the first normal distribution\n",
|
||||
"mu1, sigma1 = 3, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"data = np.random.normal(mu1, sigma1, 1000)\n",
|
||||
"\n",
|
||||
"# Create a histogram of the data\n",
|
||||
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
|
||||
"\n",
|
||||
"# Add a graph of the first normal distribution used to generate the data\n",
|
||||
"xmin, xmax = -1, 5\n",
|
||||
"x = np.linspace(xmin, xmax, 100)\n",
|
||||
"p1 = stats.norm.pdf(x, mu1, sigma1)\n",
|
||||
"plt.plot(x, p1, 'k', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
|
||||
"mu2, sigma2 = 1, 0.4\n",
|
||||
"p2 = stats.norm.pdf(x, mu2, sigma2)\n",
|
||||
"plt.plot(x, p2, 'r', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add vertical lines at x = 1 and x = 3 with labels\n",
|
||||
"plt.axvline(x=1, color='b', linestyle='--', label='mu theoretical distribution')\n",
|
||||
"plt.axvline(x=3, color='m', linestyle='--', label='mu in sample')\n",
|
||||
"\n",
|
||||
"# Calculate the 95% quantile for the second normal distribution\n",
|
||||
"quantile_95 = stats.norm.ppf(0.95, mu2, sigma2)\n",
|
||||
"\n",
|
||||
"# Plot the hatched area under the red curve for the 95% quantile and above\n",
|
||||
"x_fill = np.linspace(quantile_95, xmax, 100)\n",
|
||||
"p_fill = stats.norm.pdf(x_fill, mu2, sigma2)\n",
|
||||
"plt.fill_between(x_fill, p_fill, color='blue', alpha=0.5, hatch='/')\n",
|
||||
"\n",
|
||||
"title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu1, sigma1, mu2, sigma2)\n",
|
||||
"plt.title(title)\n",
|
||||
"plt.xlim(xmin, xmax)\n",
|
||||
"plt.legend()\n",
|
||||
"\n",
|
||||
"# Show the plot\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 38,
|
||||
"id": "665a50fe-8404-4db3-b412-aec36d81016e",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"T-statistic: 98.77295966495895, P-value: 0.0\n",
|
||||
"The means of the two theoretical distributions are statistically different (reject H0).\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Define the parameters for the two normal distributions\n",
|
||||
"mu1, sigma1 = 3, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"mu2, sigma2 = 1, 0.4 # mean and standard deviation for the second normal distribution\n",
|
||||
"\n",
|
||||
"# Perform a t-test to compare the means of the two theoretical distributions\n",
|
||||
"# Since we don't have samples, we use the means and standard deviations directly\n",
|
||||
"# Calculate the t-statistic and p-value manually\n",
|
||||
"n1 = n2 = 1000 # sample sizes (assuming large sample sizes for theoretical distributions)\n",
|
||||
"se = ((sigma1**2 / n1) + (sigma2**2 / n2))**0.5 # standard error\n",
|
||||
"t_statistic = (mu1 - mu2) / se # t-statistic\n",
|
||||
"df = n1 + n2 - 2 # degrees of freedom\n",
|
||||
"p_value = stats.t.sf(abs(t_statistic), df) * 2 # two-tailed p-value\n",
|
||||
"\n",
|
||||
"# Print the t-test result\n",
|
||||
"print(f\"T-statistic: {t_statistic}, P-value: {p_value}\")\n",
|
||||
"\n",
|
||||
"# Interpret the result\n",
|
||||
"alpha = 0.05\n",
|
||||
"if p_value < alpha:\n",
|
||||
" print(\"The means of the two theoretical distributions are statistically different (reject H0).\")\n",
|
||||
"else:\n",
|
||||
" print(\"The means of the two theoretical distributions are not statistically different (fail to reject H0).\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "63072dac-c7e6-4237-b5cc-b9832a2c7707",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now with narrowed mu"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 25,
|
||||
"id": "aa9ea3d5-3559-4fd7-a2c7-f041486c090e",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Generate 1000 random numbers from the first normal distribution\n",
|
||||
"mu1, sigma1 = 1.8, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"data = np.random.normal(mu1, sigma1, 1000)\n",
|
||||
"\n",
|
||||
"# Create a histogram of the data\n",
|
||||
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
|
||||
"\n",
|
||||
"# Add a graph of the first normal distribution used to generate the data\n",
|
||||
"xmin, xmax = -1, 5\n",
|
||||
"x = np.linspace(xmin, xmax, 100)\n",
|
||||
"p1 = stats.norm.pdf(x, mu1, sigma1)\n",
|
||||
"plt.plot(x, p1, 'k', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
|
||||
"mu2, sigma2 = 1, 0.4\n",
|
||||
"p2 = stats.norm.pdf(x, mu2, sigma2)\n",
|
||||
"plt.plot(x, p2, 'r', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add vertical lines at x = 1 and x = 3 with labels\n",
|
||||
"plt.axvline(x=mu2, color='b', linestyle='--', label='mu theoretical distribution')\n",
|
||||
"plt.axvline(x=mu1, color='m', linestyle='--', label='mu in sample')\n",
|
||||
"\n",
|
||||
"# Calculate the 95% quantile for the second normal distribution\n",
|
||||
"quantile_95 = stats.norm.ppf(0.95, mu2, sigma2)\n",
|
||||
"\n",
|
||||
"# Plot the hatched area under the red curve for the 95% quantile and above\n",
|
||||
"x_fill = np.linspace(quantile_95, xmax, 100)\n",
|
||||
"p_fill = stats.norm.pdf(x_fill, mu2, sigma2)\n",
|
||||
"plt.fill_between(x_fill, p_fill, color='blue', alpha=0.5, hatch='/')\n",
|
||||
"\n",
|
||||
"title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu1, sigma1, mu2, sigma2)\n",
|
||||
"plt.title(title)\n",
|
||||
"plt.xlim(xmin, xmax)\n",
|
||||
"plt.legend()\n",
|
||||
"\n",
|
||||
"# Show the plot\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 39,
|
||||
"id": "0db5895c-28ab-4617-ad67-7ad8311d0341",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"T-statistic: 39.50918386598358, P-value: 8.942863909656787e-253\n",
|
||||
"The means of the two theoretical distributions are statistically different (reject H0).\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Define the parameters for the two normal distributions\n",
|
||||
"mu1, sigma1 = 1.8, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"mu2, sigma2 = 1, 0.4 # mean and standard deviation for the second normal distribution\n",
|
||||
"\n",
|
||||
"# Perform a t-test to compare the means of the two theoretical distributions\n",
|
||||
"# Since we don't have samples, we use the means and standard deviations directly\n",
|
||||
"# Calculate the t-statistic and p-value manually\n",
|
||||
"n1 = n2 = 1000 # sample sizes (assuming large sample sizes for theoretical distributions)\n",
|
||||
"se = ((sigma1**2 / n1) + (sigma2**2 / n2))**0.5 # standard error\n",
|
||||
"t_statistic = (mu1 - mu2) / se # t-statistic\n",
|
||||
"df = n1 + n2 - 2 # degrees of freedom\n",
|
||||
"p_value = stats.t.sf(abs(t_statistic), df) * 2 # two-tailed p-value\n",
|
||||
"\n",
|
||||
"# Print the t-test result\n",
|
||||
"print(f\"T-statistic: {t_statistic}, P-value: {p_value}\")\n",
|
||||
"\n",
|
||||
"# Interpret the result\n",
|
||||
"alpha = 0.05\n",
|
||||
"if p_value < alpha:\n",
|
||||
" print(\"The means of the two theoretical distributions are statistically different (reject H0).\")\n",
|
||||
"else:\n",
|
||||
" print(\"The means of the two theoretical distributions are not statistically different (fail to reject H0).\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 51,
|
||||
"id": "ab987426-9a37-4c22-b561-d2870b856071",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Generate 1000 random numbers from the first normal distribution\n",
|
||||
"mu1, sigma1 = 1.03, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"data = np.random.normal(mu1, sigma1, 1000)\n",
|
||||
"\n",
|
||||
"# Create a histogram of the data\n",
|
||||
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
|
||||
"\n",
|
||||
"# Add a graph of the first normal distribution used to generate the data\n",
|
||||
"xmin, xmax = -1, 5\n",
|
||||
"x = np.linspace(xmin, xmax, 100)\n",
|
||||
"p1 = stats.norm.pdf(x, mu1, sigma1)\n",
|
||||
"plt.plot(x, p1, 'k', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
|
||||
"mu2, sigma2 = 1, 0.4\n",
|
||||
"p2 = stats.norm.pdf(x, mu2, sigma2)\n",
|
||||
"plt.plot(x, p2, 'r', linewidth=2)\n",
|
||||
"\n",
|
||||
"# Add vertical lines at x = 1 and x = 3 with labels\n",
|
||||
"plt.axvline(x=mu2, color='b', linestyle='--', label='mu theoretical distribution')\n",
|
||||
"plt.axvline(x=mu1, color='m', linestyle='--', label='mu in sample')\n",
|
||||
"\n",
|
||||
"# Calculate the 95% quantile for the second normal distribution\n",
|
||||
"quantile_95 = stats.norm.ppf(0.95, mu2, sigma2)\n",
|
||||
"\n",
|
||||
"# Plot the hatched area under the red curve for the 95% quantile and above\n",
|
||||
"x_fill = np.linspace(quantile_95, xmax, 100)\n",
|
||||
"p_fill = stats.norm.pdf(x_fill, mu2, sigma2)\n",
|
||||
"plt.fill_between(x_fill, p_fill, color='blue', alpha=0.5, hatch='/')\n",
|
||||
"\n",
|
||||
"title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu1, sigma1, mu2, sigma2)\n",
|
||||
"plt.title(title)\n",
|
||||
"plt.xlim(xmin, xmax)\n",
|
||||
"plt.legend()\n",
|
||||
"\n",
|
||||
"# Show the plot\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 50,
|
||||
"id": "9c019dea-d687-4dc1-b48f-fe629e54172e",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"T-statistic: 1.4815943949743855, P-value: 0.13860593668122914\n",
|
||||
"The means of the two theoretical distributions are not statistically different (fail to reject H0).\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Define the parameters for the two normal distributions\n",
|
||||
"mu1, sigma1 = 1.03, 0.5 # mean and standard deviation for the first normal distribution\n",
|
||||
"mu2, sigma2 = 1, 0.4 # mean and standard deviation for the second normal distribution\n",
|
||||
"\n",
|
||||
"# Perform a t-test to compare the means of the two theoretical distributions\n",
|
||||
"# Since we don't have samples, we use the means and standard deviations directly\n",
|
||||
"# Calculate the t-statistic and p-value manually\n",
|
||||
"n1 = n2 = 1000 # sample sizes (assuming large sample sizes for theoretical distributions)\n",
|
||||
"se = ((sigma1**2 / n1) + (sigma2**2 / n2))**0.5 # standard error\n",
|
||||
"t_statistic = (mu1 - mu2) / se # t-statistic\n",
|
||||
"df = n1 + n2 - 2 # degrees of freedom\n",
|
||||
"p_value = stats.t.sf(abs(t_statistic), df) * 2 # two-tailed p-value\n",
|
||||
"\n",
|
||||
"# Print the t-test result\n",
|
||||
"print(f\"T-statistic: {t_statistic}, P-value: {p_value}\")\n",
|
||||
"\n",
|
||||
"# Interpret the result\n",
|
||||
"alpha = 0.05\n",
|
||||
"if p_value < alpha:\n",
|
||||
" print(\"The means of the two theoretical distributions are statistically different (reject H0).\")\n",
|
||||
"else:\n",
|
||||
" print(\"The means of the two theoretical distributions are not statistically different (fail to reject H0).\")"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.10.12"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Reference in new issue
Block a user