Created:August 15, 2024
This calculator computes the probability of a Student's t-distribution based on the specified degrees of freedom (ν) and comparison type. It also generates a distribution chart to visualize the results.
Calculator
Parameters
Distribution Chart
Click Calculate to view the distribution chart
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Student's t-Distribution
Definition: The Student's t-distribution is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.
Formula:The probability density function (PDF) is given by:Where:
Properties
- Mean: for
- Variance: for
- Key Characteristics:
- Bell-shaped and symmetric about 0
- More peaked and heavier tails than normal distribution
- Approaches normal distribution as ν increases
- Support is all real numbers
How to Calculate t-Distribution in R
R
library(tidyverse)
df <- 10 # degrees of freedom
x1 <- -2
x2 <- 2
# P(x1 < X < x2)
prob <- pt(x2, df = df) - pt(x1, df = df)
print(str_glue("P({x1} < X < {x2}) = {round(prob, 4)}"))
# plot
x <- seq(-4, 4, length.out = 1000)
density <- dt(x, df = df)
df_plot <- tibble(x = x, density = density)
ggplot(df_plot, aes(x = x, y = density)) +
geom_line(color = "blue") +
geom_area(data = subset(df_plot, x >= x1 & x <= x2),
aes(x = x, y = density),
fill = "blue",
alpha = 0.2) +
labs(title = str_glue("Student's t-Distribution (df = {df})"),
x = "t",
y = "Probability Density") +
geom_vline(xintercept = 0,
linetype = "dashed",
color = "gray50") +
theme_minimal()How to Calculate t-Distribution in Python
Python
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
import seaborn as sns
df = 10 # degrees of freedom
x1 = -2
x2 = 2
# P(x1 < X < x2)
prob = stats.t.cdf(x2, df) - stats.t.cdf(x1, df)
print(f"P({x1} < X < {x2}) = {prob:.4f}")
# plot
x = np.linspace(-4, 4, 1000)
pdf = stats.t.pdf(x, df)
plt.figure(figsize=(10, 6))
plt.plot(x, pdf, 'blue', label='PDF')
x_shade = x[(x >= x1) & (x <= x2)]
pdf_shade = stats.t.pdf(x_shade, df)
plt.fill_between(x_shade, pdf_shade, alpha=0.2, color='blue')
plt.axvline(x=0, color='gray', linestyle='--', alpha=0.5)
plt.title(f"Student's t-Distribution (df = {df})")
plt.xlabel('t')
plt.ylabel('Probability Density')
plt.grid(True, alpha=0.3)
plt.show()