Lernen Working with Probability Distributions

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Understanding probability distributions is essential for analyzing and interpreting data in statistics. A probability distribution describes how the values of a random variable are distributed – the likelihood of each possible outcome. Three of the most common distributions you will encounter are the normal, binomial, and uniform distributions.

The normal distribution is a continuous, bell-shaped curve that is symmetric around its mean. It is defined by two parameters: the mean ( $μ$ ) and the standard deviation ( $σ$ ). Many natural phenomena, such as human heights or measurement errors, follow a normal distribution. The mean determines the center of the distribution, while the standard deviation controls the spread.

The binomial distribution is a discrete distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by two parameters: the number of trials ( $n$ ) and the probability of success ( $p$ ). Tossing a coin several times and counting the number of heads is a classic example of a binomial distribution.

The uniform distribution is the simplest probability distribution, where all outcomes in a given range are equally likely. The continuous version is defined by two parameters: the lower bound ( $a$ ) and the upper bound ( $b$ ). Rolling a fair die or generating a random number between 0 and 1 are examples of uniform distributions.

Each of these distributions helps you model different types of random processes and understand the likelihood of various outcomes.


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import numpy as np
import matplotlib.pyplot as plt

# Set random seed for reproducibility
np.random.seed(42)

# Generate samples
normal_samples = np.random.normal(loc=0, scale=1, size=1000)      # mean=0, std=1
binomial_samples = np.random.binomial(n=10, p=0.5, size=1000)     # 10 trials, p=0.5
uniform_samples = np.random.uniform(low=0, high=1, size=1000)     # between 0 and 1

# Plotting
fig, axs = plt.subplots(1, 3, figsize=(15, 4))

# Normal distribution
axs[0].hist(normal_samples, bins=30, color='skyblue', edgecolor='black', density=True)
axs[0].set_title('Normal Distribution\n(mean=0, std=1)')
axs[0].set_xlabel('Value')
axs[0].set_ylabel('Density')

# Binomial distribution
axs[1].hist(binomial_samples, bins=np.arange(-0.5, 11.5, 1), color='salmon', edgecolor='black', density=True)
axs[1].set_title('Binomial Distribution\n(n=10, p=0.5)')
axs[1].set_xlabel('Number of Successes')
axs[1].set_ylabel('Probability')

# Uniform distribution
axs[2].hist(uniform_samples, bins=30, color='lightgreen', edgecolor='black', density=True)
axs[2].set_title('Uniform Distribution\n(low=0, high=1)')
axs[2].set_xlabel('Value')
axs[2].set_ylabel('Density')

plt.tight_layout()
plt.show()

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