Descriptive Statistics
Normal distribution \(\xi \sim \mathcal{N}(\mu, \sigma^2)\) is a continuous distribution in which the density decreases with increasing distance from \(\mu\) exponentially.
\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]
where:
# Ініціалізація
set.seed(2024)
# Параметри розподілу
mu <- 0
sigma <- 2
x <- seq(-8, 8, length.out = 1000) # рівновіддалені 1000 точок від -8 до 8
# Розрахунок щільності, кумулятивної функції розподілу та вибірки
pdf <- dnorm(x, mean = mu, sd = sigma)
cdf <- pnorm(x, mean = mu, sd = sigma)
sample <- rnorm(10000, mean = mu, sd = sigma)
# Візуалізація
library(ggplot2)
library(gridExtra)
# Щільність
p1 <- ggplot() +
geom_histogram(aes(x = sample, y = ..density..), fill = "steelblue", alpha = 0.6) +
geom_line(aes(x = x, y = pdf), color = "darkred", size = 1) +
labs(title = "Density visualisation", x = "Значення", y = "Щільність") +
theme_minimal()
# Кумулятивна функція
p2 <- ggplot() +
geom_line(aes(x = x, y = cdf), color = "darkgreen", size = 1) +
labs(title = "Cumulative distribution function", x = "Значення", y = "CDF") +
theme_minimal()
# Показати графіки
grid.arrange(p1, p2, ncol = 2)
# Генерація вибірок
set.seed(2024)
first_sample <- rnorm(10000, mean = first_dist$mean, sd = first_dist$sd)
second_sample <- rnorm(10000, mean = second_dist$mean, sd = second_dist$sd)
sum_sample <- first_sample + second_sample
x <- seq(-8, 12, length.out = 1000)
checking_sum_pdf <- dnorm(x, mean = checking_sum_distr$mean, sd = checking_sum_distr$sd)
ggplot() +
geom_histogram(aes(x = sum_sample, y = ..density..), fill = "steelblue", alpha = 0.6, bins = 40) +
geom_line(aes(x = x, y = checking_sum_pdf), color = "darkred", size = 1) +
geom_density(aes(x = sum_sample), color = "darkblue", size = 1) +
labs(
title = "Перевірка розподілів",
x = "X",
y = "Щільність"
) +
theme_minimal()
binom_generator <- function(p, n, size) {
rbinom(size, size = n, prob = p)
}
p <- 0.01
n <- 20
size <- 5000
visualize_CLT(
sample_generator = function() binom_generator(p = p, n = n, size = size),
expected_value = p * n,
variance = n * p * (1 - p)
)
binom_generator <- function(p, n, size) {
rbinom(size, size = n, prob = p)
}
p <- 0.01
n <- 20
size <- 40
visualize_CLT(
sample_generator = function() binom_generator(p = p, n = n, size = size),
expected_value = p * n,
variance = n * p * (1 - p)
)
expon_generator <- function(expected_value, size) {
rexp(size, rate = 1 / expected_value)
}
expected_value <- 24
size <- 400
visualize_CLT(
sample_generator = function() expon_generator(expected_value = expected_value, size = size),
expected_value = expected_value,
variance = expected_value^2
)
cmp_pvalue_binom_and_norm <- function(n, mu0, t, add_to_x = 0) {
# Parameters
sum_mu <- n * mu0
sum_variance <- n * mu0 * (1 - mu0)
sum_std <- sqrt(sum_variance)
x_axis <- seq(0, n, length.out = 1000)
dots_to_show <- seq(0, n, by = 1)
# Plot
ggplot() +
# Binomial p-value
geom_point(aes(x = dots_to_show, y = 1 - pbinom(dots_to_show - 1, size = n, prob = mu0)),
color = "slategray",
size = 3) +
geom_point(aes(x = t, y = 1 - pbinom(t - 1, size = n, prob = mu0), color = "slategray"),
size = 5) +
# Normal p-value
geom_line(aes(x = x_axis,
y = 1 - pnorm(x_axis + add_to_x, mean = sum_mu, sd = sum_std)),
color = red,
alpha = 0.5) +
geom_point(aes(x = t, y = 1 - pnorm(t + add_to_x, mean = sum_mu, sd = sum_std), color = "red"),
size = 5) +
labs(title = "Comparison of p-value: binomial and normal",
x = "t", y = "p-value") +
theme_minimal() +
scale_colour_manual(name = "p-value", values = c('red' = red, slategray = "slategray"),
labels = c("Normal distribution: t = 15", "Binomial distribution: t = 15"))
}
cmp_pvalue_binom_and_norm(30, 0.5, 15)\[F_{\text{new}}(x) = F_{\text{old}}(x - 0.5)\]
cmp_pvalue_binom_and_norm(30, 0.5, 15, add_to_x=-0.5)
