What is Student’s t-distribution?

The t-distribution, also known as the Student’s t-distribution, is a probability distribution that is widely used in statistical analysis, particularly in hypothesis testing and in constructing confidence intervals when the sample size is small or the population variance is unknown.

In R, the t-distribution can be worked with using the following functions:

1. dt: Probability Density Function (PDF) of the t-distribution
2. pt: Cumulative Distribution Function (CDF) of the t-distribution
3. qt: Quantile Function (inverse of the CDF) of the t-distribution
4. rt: Random number generation from the t-distribution

To use these functions, you need to specify the degrees of freedom (df) for the t-distribution. Here’s a quick overview of each function with examples:

1. dt(x, df):

This function computes the PDF of the t-distribution. It takes two arguments: ‘x’, the value at which the PDF is evaluated, and ‘df’, the degrees of freedom.

 

x <- seq(-5, 5, length.out = 100)
df <- 10
pdf <- dt(x, df)
plot(x, pdf, type="l", 
main="PDF of t-distribution with 10 degrees of freedom")

2. pt(x, df):

This function computes the CDF of the t-distribution. It also requires two arguments: ‘x’ and ‘df’.

x <- seq(-5, 5, length.out = 100)
df <- 10
cdf <- pt(x, df)
plot(x, cdf, 
type="l", main="CDF of t-distribution with 10 degrees of freedom")

3. qt(p, df):

This function computes the quantiles (inverse of the CDF) of the t-distribution. It takes two arguments: ‘p’, the probability for which you want to find the corresponding quantile, and ‘df’, the degrees of freedom.

p <- c(0.025, 0.975)
df <- 10
quantiles <- qt(p, df)
print(quantiles)

4. rt(n, df):

This function generates random numbers from the t-distribution. It takes two arguments: ‘n’, the number of random numbers you want to generate, and ‘df’, the degrees of freedom.

n <- 100
df <- 10
random_numbers <- rt(n, df)
hist(random_numbers, 
main="Random numbers from t-distribution with 10 degrees of freedom")

Remember to adjust the degrees of freedom (df) according to your specific problem or dataset.