Lesson 5

Welcome to our vivid exploration of **probability distributions**! In this lesson, we're going to delve into different types of probability distributions, specifically the *Uniform*, *Normal* and *Binominal* distributions. We will leverage R's libraries to create visualizations of these distributions.

Probability quantifies the likelihood of the occurrence of an event among all potential outcomes. For instance, if we toss a coin, the likelihood of obtaining a head is 50%, or 0.5. Essentially, probability distributions map out each outcome of a random variable along with its corresponding probability.

To visualize the distributions under study, we will employ the power of the `ggplot2`

library in R. At this stage, you can regard `ggplot2`

as an exceptional tool that aids us in our learning. Remember, our primary focus lies in exploring statistical distributions. There is no need to understand precisely how to use this library, but you will be provided with a fully-working code for this lesson and the following practices. We will cover the details of data visualization in R in one of the following courses.

Imagine a situation where all outcomes are equally likely to occur. This scenario can be depicted by a Uniform Distribution. For instance, if we pick a suit from a deck of cards, the probabilities of getting a heart, club, diamond, or spade are equal. Let's generate and plot a Uniform Distribution using `runif()`

and `ggplot2`

.

R`1library(ggplot2) 2set.seed(123) 3 4# Generate random numbers uniformly distributed between -1 and 1 5uniform_data <- runif(1000, min = -1, max = 1) 6 7# Plot a Histogram of the distribution 8plot <- ggplot() + 9 geom_histogram(aes(uniform_data), bins = 20, fill = 'dodgerblue3', alpha = 0.7) + 10 labs(title = "Uniform Distribution")`

Here, `runif(1000, min = -1, max = 1)`

generates 1000 random numbers uniformly distributed between -1 and 1. The `geom_histogram`

function constructs a histogram of the distribution.

Let's shift our focus to the **Normal Distribution**, which is a statistical function characterized by a bell-shaped curve and used extensively in statistical analysis. A key feature of the Normal Distribution lies in its definition via just two parameters: the mean (average) and the standard deviation (spread). Let's simulate and plot a Normal Distribution:

R`1# Generate Normal Distribution data 2normal_data <- rnorm(1000, mean = 0, sd = 1) 3 4# Plot a Histogram of the distribution 5plot <- ggplot() + 6 geom_histogram(aes(normal_data), bins = 20, fill = 'dodgerblue3', alpha = 0.7) + 7 labs(title = "Normal Distribution")`

The function `rnorm(1000, mean = 0, sd = 1)`

generates 1000 data points conforming to a Normal Distribution with a mean of 0 and a standard deviation of 1.

Following our journey through Uniform and Normal distributions, let's examine the **Binomial Distribution**. This type of distribution is particularly useful when dealing with scenarios with two possible outcomes (success or failure) in a series of independent trials. A classic example of binomial distribution is flipping a coin several times and counting the number of heads (or tails).

We can use the `rbinom()`

function to simulate and visualize a Binomial Distribution in R. Let's perform an experiment where we flip a coin 10 times, and we want to know the distribution of getting heads in those flips over 1000 trials.

R`1# Generate Binomial Distribution data 2binomial_data <- rbinom(1000, size = 10, prob = 0.5) # size = number of trials, prob = probability of success 3 4# Plot a Histogram of the distribution 5plot <- ggplot() + 6 geom_histogram(aes(binomial_data), bins = 20, fill = 'dodgerblue3', alpha = 0.7) + 7 labs(title = "Binomial Distribution")`

In this code, `rbinom(1000, size = 10, prob = 0.5)`

generates data from 1000 experiments where each experiment consists of flipping a coin 10 times (`size = 10`

) with the probability of getting a head in any single trial being 50% (`prob = 0.5`

). The histogram visualizes the distribution of the number of heads out of 10 flips over the 1000 trials.

Analyzing the histogram of the Binomial Distribution provides insights into the likelihood of achieving a certain number of successes (e.g., heads in our example) in a set number of trials. For instance, you might observe that obtaining 5 heads in 10 flips is the most common outcome, aligning with our expectations given the 50/50 nature of a coin flip.

Understanding the Binomial Distribution equips us with the tools to tackle questions about probability in binary scenarios, making it an integral part of our statistical toolkit along with Uniform and Normal distributions.

Calculating metrics such as mean (average), variance (spread), skewness (asymmetry), and kurtosis (sharpness of the curve's peak) can help us understand our distributions better. Let's perform these calculations in R — we are already equipped with the necessary knowledge!

R`1install.packages("moments") 2library(moments) 3 4normal_data <- rnorm(1000, mean = 0, sd = 1) 5 6# Calculate properties 7mean_val = mean(normal_data) # Mean 8var_val = var(normal_data) # Variance 9skew_val = skewness(normal_data) # Skewness 10kurt_val = kurtosis(normal_data) # Kurtosis 11 12# Print the results 13cat("Mean: ", mean_val, "\n") 14cat("Variance: ", var_val, "\n") 15cat("Skewness: ", skew_val, "\n") 16cat("Kurtosis: ", kurt_val, "\n")`

Fantastic! You've just gained a comprehensive understanding of probability, Uniform and Normal distributions. Not only did you learn the theory, but you also carried out simulations, created visualizations, and interpreted these probability distributions using R. Now, it's time to put theory into practice with hands-on exercises. Enhancing your understanding and skillset in data analytics through the application of theoretical knowledge can be significantly advantageous. Let's keep the momentum going!