Probability theory and random variables

ML assignment blog 1
Machine Learning
Published

December 1, 2023

Probability theory is a branch of mathematics that deals with the study of uncertainty and randomness. It provides a framework for quantifying and reasoning about uncertainty, allowing us to model and analyze various phenomena that involve chance or randomness. The central concept in probability theory is the probability, which represents the likelihood of different outcomes in a random experiment.

1. Required packages and probability density

#install.packages("palmerpenguins")

library(tidyr)
library(palmerpenguins)
library(ggplot2)

data("penguins")

str(penguins)
tibble [344 × 8] (S3: tbl_df/tbl/data.frame)
 $ species          : Factor w/ 3 levels "Adelie","Chinstrap",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ island           : Factor w/ 3 levels "Biscoe","Dream",..: 3 3 3 3 3 3 3 3 3 3 ...
 $ bill_length_mm   : num [1:344] 39.1 39.5 40.3 NA 36.7 39.3 38.9 39.2 34.1 42 ...
 $ bill_depth_mm    : num [1:344] 18.7 17.4 18 NA 19.3 20.6 17.8 19.6 18.1 20.2 ...
 $ flipper_length_mm: int [1:344] 181 186 195 NA 193 190 181 195 193 190 ...
 $ body_mass_g      : int [1:344] 3750 3800 3250 NA 3450 3650 3625 4675 3475 4250 ...
 $ sex              : Factor w/ 2 levels "female","male": 2 1 1 NA 1 2 1 2 NA NA ...
 $ year             : int [1:344] 2007 2007 2007 2007 2007 2007 2007 2007 2007 2007 ...
summary(penguins)
      species          island    bill_length_mm  bill_depth_mm  
 Adelie   :152   Biscoe   :168   Min.   :32.10   Min.   :13.10  
 Chinstrap: 68   Dream    :124   1st Qu.:39.23   1st Qu.:15.60  
 Gentoo   :124   Torgersen: 52   Median :44.45   Median :17.30  
                                 Mean   :43.92   Mean   :17.15  
                                 3rd Qu.:48.50   3rd Qu.:18.70  
                                 Max.   :59.60   Max.   :21.50  
                                 NA's   :2       NA's   :2      
 flipper_length_mm  body_mass_g       sex           year     
 Min.   :172.0     Min.   :2700   female:165   Min.   :2007  
 1st Qu.:190.0     1st Qu.:3550   male  :168   1st Qu.:2007  
 Median :197.0     Median :4050   NA's  : 11   Median :2008  
 Mean   :200.9     Mean   :4202                Mean   :2008  
 3rd Qu.:213.0     3rd Qu.:4750                3rd Qu.:2009  
 Max.   :231.0     Max.   :6300                Max.   :2009  
 NA's   :2         NA's   :2                                 
unique(penguins$species)
[1] Adelie    Gentoo    Chinstrap
Levels: Adelie Chinstrap Gentoo
flipper_length = penguins$flipper_length_mm

summary(flipper_length)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
  172.0   190.0   197.0   200.9   213.0   231.0       2 
flip_len = na.omit(flipper_length)

summary(flip_len)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  172.0   190.0   197.0   200.9   213.0   231.0 
# Select one continuous variables for probability density plots
selected_vars <- c(penguins$flipper_length_mm)
                   
selected_vars2 <- c(penguins$bill_length_mm)                   
                   
selected_vars3 <- c(penguins$body_mass_g)   

# Create probability density plots for each variable by species

penguin_density_plots <- ggplot(penguins, aes(x = selected_vars, fill = species)) +
  geom_density(alpha = 0.7) +
  facet_wrap(~species, scales = "free") +
  labs(title = paste("Probability Density Plots of", selected_vars, "by Species"), x = "flipper length", y = "Density")


print(penguin_density_plots)

penguin_density_plots2 <- ggplot(penguins, aes(x = selected_vars2, fill = species)) +
  geom_density(alpha = 0.7) +
  facet_wrap(~species, scales = "free") +
  labs(title = paste("Probability Density Plots of", selected_vars2, "by Species"), x = "bill length", y = "Density")


print(penguin_density_plots2)
Warning: Removed 2 rows containing non-finite values (`stat_density()`).

penguin_density_plots3 <- ggplot(penguins, aes(x = selected_vars3, fill = species)) +
  geom_density(alpha = 0.7) +
  facet_wrap(~species, scales = "free") +
  labs(title = paste("Probability Density Plots of", selected_vars3, "by Species"), x = "Body mass", y = "Density")


print(penguin_density_plots3)
Warning: Removed 2 rows containing non-finite values (`stat_density()`).

2. Conditional probability

#conditional probabilities

# Create a contingency table for species and island
contingency_table_island <- table(penguins$species, penguins$island)

# Calculate conditional probabilities given the island
conditional_probabilities_island <- prop.table(contingency_table_island, margin = 1)

# Create a contingency table for species and sex
contingency_table_sex <- table(penguins$species, penguins$sex)

# Calculate conditional probabilities given the sex
conditional_probabilities_sex <- prop.table(contingency_table_sex, margin = 1)

# Print the contingency tables
print("Contingency Table for Species and Island:")
[1] "Contingency Table for Species and Island:"
print(contingency_table_island)
           
            Biscoe Dream Torgersen
  Adelie        44    56        52
  Chinstrap      0    68         0
  Gentoo       124     0         0
print("Contingency Table for Species and Sex:")
[1] "Contingency Table for Species and Sex:"
print(contingency_table_sex)
           
            female male
  Adelie        73   73
  Chinstrap     34   34
  Gentoo        58   61
# Print conditional probabilities
print("Conditional Probabilities Given Island:")
[1] "Conditional Probabilities Given Island:"
print(conditional_probabilities_island)
           
               Biscoe     Dream Torgersen
  Adelie    0.2894737 0.3684211 0.3421053
  Chinstrap 0.0000000 1.0000000 0.0000000
  Gentoo    1.0000000 0.0000000 0.0000000
print("Conditional Probabilities Given Sex:")
[1] "Conditional Probabilities Given Sex:"
print(conditional_probabilities_sex)
           
              female     male
  Adelie    0.500000 0.500000
  Chinstrap 0.500000 0.500000
  Gentoo    0.487395 0.512605

3. Random variables

A random variable is a mathematical function that assigns a numerical value to each outcome in the sample space of a random experiment. It serves as a way to quantify and analyze the variability and uncertainty associated with random processes. Random variables can be classified as either discrete or continuous.

# Simulate rolling a fair six-sided die
die_outcomes <- 1:6

# Define random variables X and Y
X <- sample(die_outcomes, 1, replace = TRUE)
Y <- sample(die_outcomes, 1, replace = TRUE)

# Print the outcomes of X and Y
cat("Outcome of X:", X, "\n")
Outcome of X: 5 
cat("Outcome of Y:", Y, "\n")
Outcome of Y: 2 
# Calculate the sum Z
Z <- X + Y
cat("Sum Z:", Z, "\n")
Sum Z: 7 
# Simulate rolling the dice multiple times to estimate probabilities
num_simulations <- 10000

# Simulate X and Y
simulated_X <- sample(die_outcomes, num_simulations, replace = TRUE)
simulated_Y <- sample(die_outcomes, num_simulations, replace = TRUE)

# Calculate the sum Z for each simulation
simulated_Z <- simulated_X + simulated_Y

# Calculate the empirical probabilities
empirical_probabilities <- table(simulated_Z) / num_simulations

# Print the empirical probabilities
cat("Empirical Probabilities for Z:\n")
Empirical Probabilities for Z:
print(empirical_probabilities)
simulated_Z
     2      3      4      5      6      7      8      9     10     11     12 
0.0294 0.0550 0.0895 0.1085 0.1344 0.1649 0.1401 0.1132 0.0839 0.0542 0.0269 
# Plot the empirical probabilities
barplot(empirical_probabilities, names.arg = 2:12, col = "skyblue", main = "Empirical Probabilities of Dice Sum", xlab = "Sum", ylab = "Probability")