# R for Data Analytics – Distribution Functions

## Introduction

In data analytics, understanding the distribution of your data is essential for making informed decisions, testing hypotheses, and estimating parameters. R, a powerful programming language for statistical computing and data analysis, provides a comprehensive set of built-in functions for working with probability distributions. This article will discuss some of the most common probability distributions and their corresponding functions in R, as well as demonstrate how to use these functions for data analytics tasks.

## Common Probability Distributions in R

R includes functions for numerous probability distributions, such as the normal, binomial, Poisson, exponential, and gamma distributions. For each distribution, R provides four main types of functions:

a. Density (d): Calculates the probability density function (PDF) or probability mass function (PMF) for discrete distributions. b. Cumulative (p): Computes the cumulative distribution function (CDF). c. Quantile (q): Finds the quantiles or percentiles of the distribution. d. Random (r): Generates random numbers from the specified distribution.

These functions are prefixed with the first letter of the distribution name (e.g., ‘n’ for normal, ‘b’ for binomial) and the type of function (d, p, q, or r). For example, the density function for the normal distribution is dnorm(), while the quantile function for the binomial distribution is qbinom().

### Normal Distribution Functions

The normal (Gaussian) distribution is a continuous distribution characterized by its bell-shaped curve, with a mean (μ) and standard deviation (σ) as its parameters. R provides the following functions for working with the normal distribution:

- dnorm(x, mean = 0, sd = 1): Calculates the PDF for given values of x, with optional mean and standard deviation arguments.
- pnorm(q, mean = 0, sd = 1): Computes the CDF for given quantiles q, with optional mean and standard deviation arguments.
- qnorm(p, mean = 0, sd = 1): Finds the quantiles corresponding to given probabilities p, with optional mean and standard deviation arguments.
- rnorm(n, mean = 0, sd = 1): Generates n random numbers from a normal distribution with the specified mean and standard deviation.

### Binomial Distribution Functions

The binomial distribution is a discrete distribution representing the number of successes in a fixed number of Bernoulli trials, with a probability of success (p) and number of trials (n) as its parameters. R provides the following functions for working with the binomial distribution:

- dbinom(x, size, prob): Calculates the PMF for given values of x, number of trials (size), and probability of success (prob).
- pbinom(q, size, prob): Computes the CDF for given quantiles q, number of trials (size), and probability of success (prob).
- qbinom(p, size, prob): Finds the quantiles corresponding to given probabilities p, number of trials (size), and probability of success (prob).
- rbinom(n, size, prob): Generates n random numbers from a binomial distribution with the specified number of trials and probability of success.

### Poisson, Exponential, and Gamma Distribution Functions

R also provides functions for working with other common distributions, such as the Poisson, exponential, and gamma distributions. For example:

- Poisson distribution functions: dpois(), ppois(), qpois(), and rpois()
- Exponential distribution functions: dexp(), pexp(), qexp(), and rexp()
- Gamma distribution functions: dgamma(), pgamma(), qgamma(), and rgamma()

### Using Distribution Functions in Data Analytics

Distribution functions in R can be employed in various data analytics tasks, such as hypothesis testing, parameter estimation, and data simulation. Below are some examples demonstrating the use of distribution functions in R:

**a. Hypothesis Testing: One-sample t-test**

Suppose you have a sample of 30 observations with a mean of 105 and a standard deviation of 15. You want to test if the true population mean is 100. You can use the t-distribution functions to perform a one-sample t-test:

```
n <- 30
sample_mean <- 105
sample_sd <- 15
population_mean <- 100
# Calculate the t-statistic
t_statistic <- (sample_mean - population_mean) / (sample_sd / sqrt(n))
# Calculate the p-value (two-tailed test)
p_value <- 2 * (1 - pt(abs(t_statistic), df = n - 1))
# Display the test results
cat("t-statistic:", t_statistic, "\np-value:", p_value)
```

**b. Parameter Estimation: Confidence Interval for a Population Mean**

Using the same sample from the previous example, you can calculate the 95% confidence interval for the true population mean:

```
alpha <- 0.05
critical_value <- qt(1 - alpha / 2, df = n - 1)
margin_of_error <- critical_value * (sample_sd / sqrt(n))
confidence_interval <- c(sample_mean - margin_of_error, sample_mean + margin_of_error)
# Display the confidence interval
cat("95% Confidence Interval:", confidence_interval)
```

**c. Data Simulation: Generating Random Data**

You can use R’s random number functions to simulate data from various probability distributions. For example, you can generate 100 random observations from a normal distribution with a mean of 50 and a standard deviation of 10:

```
random_data <- rnorm(n = 100, mean = 50, sd = 10)
# Display the first 10 observations
cat("First 10 observations:", random_data[1:10])
```

## Conclusion

R’s extensive set of distribution functions makes it an invaluable tool for data analytics tasks involving probability distributions. By understanding and using these functions, you can perform hypothesis testing, estimate parameters, simulate data, and much more. With the power of R and its distribution functions, you can effectively analyze your data and make informed decisions based on statistical insights.

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# R for Data Analytics – Distribution Functions

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