LAB 6: Probability Distributions

This lab demonstrates various probability distributions including normal, uniform, binomial, Poisson, and exponential distributions. We'll visualize each distribution and understand their characteristics.

# Import necessary libraries
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# Set up the figure for multiple subplots
plt.figure(figsize=(15, 10))

# 1. Basic Probability Distribution (Random Data)
data_rand = np.random.randn(1000)
plt.subplot(2, 3, 6)
plt.hist(data_rand, bins=30, color='orange')
plt.title("Histogram of Random Data")
plt.xlabel("Value")
plt.ylabel("Frequency")

# 2. Normal Distribution (Continuous)
x_norm = np.linspace(-5, 5, 100)
y_norm = stats.norm.pdf(x_norm, 0, 1)  # mean=0, std=1
plt.subplot(2, 3, 1)
plt.plot(x_norm, y_norm)
plt.title("Standard Normal Distribution")
plt.xlabel("Value")
plt.ylabel("Probability Density")
plt.grid(True)

# 3. Uniform Distribution (Continuous)
data_uni = np.random.uniform(0, 1, 1000)
plt.subplot(2, 3, 2)
plt.hist(data_uni, bins=20, color='skyblue', edgecolor='black')
plt.title("Uniform Distribution (0 to 1)")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.grid(True)

# 4. Binomial Distribution (Discrete)
n, p = 10, 0.5  # number of trials, probability of success
x_bin = np.arange(0, n+1)
y_bin = stats.binom.pmf(x_bin, n, p)
plt.subplot(2, 3, 3)
plt.bar(x_bin, y_bin, color='salmon')
plt.title("Binomial Distribution (n=10, p=0.5)")
plt.xlabel("Number of Successes")
plt.ylabel("Probability")

# 5. Poisson Distribution (Discrete)
lambda_p = 3  # average number of events
x_poi = np.arange(0, 15)
y_poi = stats.poisson.pmf(x_poi, lambda_p)
plt.subplot(2, 3, 4)
plt.bar(x_poi, y_poi, color='lightgreen')
plt.title("Poisson Distribution (λ=3)")
plt.xlabel("Number of Events")
plt.ylabel("Probability")

# 6. Exponential Distribution (Continuous)
lambda_exp = 0.5
x_exp = np.linspace(0, 10, 100)
y_exp = stats.expon.pdf(x_exp, scale=1/lambda_exp)
plt.subplot(2, 3, 5)
plt.plot(x_exp, y_exp, color='purple')
plt.title("Exponential Distribution (λ=0.5)")
plt.xlabel("Time")
plt.ylabel("Probability Density")

plt.tight_layout()
plt.show()

A probability distribution describes the likelihood of different outcomes in a random event.

Normal Distribution

The normal (Gaussian) distribution is a symmetric bell-shaped curve used for real-valued random variables.

Uniform Distribution

The uniform distribution represents constant probability over a specified range.

Binomial Distribution

The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials.

Poisson Distribution

The Poisson distribution represents the number of events occurring in a fixed interval of time or space.

Exponential Distribution

The exponential distribution represents the time between events in a Poisson process.

Exercise 7: Probability Distributions

Write a python program for applying Probability Distribution, Normal Distribution, Uniform Distribution. Binomial Distribution, Poisson Distribution and Exponential Distribution.

The code above demonstrates the implementation as described in the exercise.

Summary of Distributions

Distribution	Type	Description
Normal	Continuous	Symmetric bell-shaped curve; used for real-valued random variables.
Uniform	Continuous	Represents constant probability over a specified range.
Binomial	Discrete	Number of successes in a fixed number of independent trials.
Poisson	Discrete	Number of events occurring in a fixed interval of time or space.
Exponential	Continuous	Represents the time between events in a Poisson process.