# ========================= 
# Basic Statistical Distributions and Data Preprocessing 
# ========================= 
# ----------------------------------- 
# 1. Basic Statistical Distributions 
# ----------------------------------- 

# Probability Distributions - Probabilities 
# Binomial Distribution 
# For example, 10 trials, 3 successes, success probability=0.5 
dbinom(x=0:10, size=10, prob=0.5) 
# Calculating probability of exactly 3 successes 
dbinom(3, 10, 0.5) 

# Probabilities for all possible outcomes 
plot(0:10, dbinom(0:10, 10, 0.5), type='h',  
     main="Binomial Distribution (n=10, p=0.5)", xlab="Number of Successes",
     ylab="Probability")


# Normal Distribution (Gaussian) 
# Typically models continuous data 
x <- seq(-4, 4, length=100) 
y <- dnorm(x, mean=0, sd=1) 
plot(x, y, type='l', main="Normal Distribution (mean=0, sigma=1)", ylab="Density", xlab="Values") 

# Geometric Distribution 
# For example, number of trials until the first success with p=0.3 
x <- 1:20 
plot(x, dgeom(x-1, prob=0.3), type='h',  
     main="Geometric Distribution", xlab="Trials until first success", ylab="Probability") 

# Estimating probabilities with the normal distribution 
# Probability that a standard normally distributed variable is within ±1 std dev 
pnorm(1, mean=0, sd=1) - pnorm(-1, mean=0, sd=1) 

# ----------------------------------- 
# 2. Plotting with Base R 
# ----------------------------------- 

# Sampling from a normal distribution 
sample_data <- rnorm(1000, mean=0, sd=1) 

# Histogram of the sample 
hist(sample_data, main="Histogram of Normal Distribution", col="orange", xlab="Values") 

# Bar plot of positive vs. negative counts 
positive_counts <- table(sample_data > 0) 
barplot(positive_counts, main="Proportion of Positive Data", col="skyblue", ylab="Count") 

# Overlay normal density on sample data plot 
plot(sample_data, main="Sample Data and Normal Density", col="purple")
hist(sample_data, probability=TRUE, main="Histogram with Normal Density", col="orange", xlab="Values")
curve(dnorm(x, mean=mean(sample_data), sd=sd(sample_data)), add=TRUE, col="red", lwd=2)



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# 3. Data Preprocessing: Handling Outliers & Missing Values 
# ----------------------------------- 

# Create sample data with missing values and outliers 
set.seed(123) 
data <- rnorm(100)  # Generate 100 random normal values 
data[sample(1:100, 10)] <- NA  # Inject missing values (NA) 
data[which(data > 3)] <- 10    # Add some outliers (values outside typical range) 

# Check for missing data 
sum(is.na(data))  # Count of missing (NA) values 

# Removing missing values 
data_no_na <- na.omit(data) 

# Find outliers based on a common rule (e.g., beyond 3 standard deviations) 
mean_data <- mean(data, na.rm=TRUE) 
sd_data <- sd(data, na.rm=TRUE) 
lower_bound <- mean_data - 3*sd_data 
upper_bound <- mean_data + 3*sd_data 

# Identify outliers 
outliers <- data[which(data < lower_bound | data > upper_bound)] 
print("Detected outliers:") 
print(outliers) 

# Visualize data with outliers 
hist(data, main="Histogram with Outliers and Missing Values", col="lightgreen", xlab="Values") 
abline(v=c(lower_bound, upper_bound), col="red", lty=2) 
legend("topright", legend=c("Outlier thresholds"), col="red", lty=2) 

# Summary of data excluding missing values 
summary(data, na.rm=TRUE)