# ==============================================================================
# --------------------------------- SEMINAR 5 ----------------------------------
# ==============================================================================


# ----------------------------------- TASK 1 -----------------------------------


# ...................................... A .....................................


# Create density plot, distribution function plot, and quantile function plot
# of random variable X with binomial distribution Bi(n = 100, p = 0.5).

# HINT: apply these functions on the values from the x-axis:
?pbinom
?dbinom
?qbinom

### DISTRIBUTION FUNCTION:

xx <- 0:100

dist.b <- pbinom(xx, size = 100, prob = 0.5)

# computes the values of distribution function for each value of vector 0:100
# of all possible outcomes (number of successes) of binomial distribution

plot(xx, dist.b, type = 's', col = "red",
     main = "Distribution", xlab = "", ylab = "")

### DENSITY FUNCTION:

xx <- 0:100

density.b <- dbinom(xx, size = 100, prob = 0.5)

plot(xx, density.b, type = 'h', col = "red",
     main = "Probability mass function", xlab = "", ylab = "")
points(xx, density.b, pch = 19, col = "darkred", cex = 0.5)

# computes the values of density function for each value of vector 0:100
# of all possible outcomes (number of successes) of binomial distribution

### QUANTILE FUNCTION:

xx <- seq(0, 1, by = 0.01)

quantile.b <- qbinom(xx, size = 100, prob = 0.5)

plot(xx, quantile.b, type = 's', col = "red",
     main = "Quantile function", xlab = "", ylab = "")

# computes the values of quantile function for each value of given vector of values
# between 0 and 1 of different probabilities


# ...................................... B .....................................


# Generate random vector of length 1000 with independent ele-
# ments, where each element X is following a binomial distribution
# Bi(100; 0.5). Produce a histogram and compare it to the
# desinty plot of the same distribution.

?rbinom

n <- rbinom(1000, size = 100, prob = 0.5) # vector of 1000 random numbers
hist(n, xlim = c(0, 100), breaks = seq(0, 1000, by = 3), freq = F, col = "salmon2")

d.b <- dbinom(0:100, size = 100, prob = 0.5) # density of binomial distribution
points(0:100, d.b, pch = 19, type = "b")


# ----------------------------------- TASK 2 -----------------------------------


# ...................................... A .....................................


# The same for Poisson distribution Po(lambda = 10)

# HINT: apply these functions on the values from the x-axis:
?ppois
?dpois
?qpois

### DISTRIBUTION FUNCTION:

dist.p <- ppois(0:30, lambda = 10)

plot(0:30, dist.p, type = 's', col = "red",
     main = "Distribution", xlab = "", ylab = "")

### DENSITY FUNCTION:

density.p <- dpois(0:30, lambda = 10)

plot(0:30, density.p, type = 'h', col = "red",
     main = "Probability mass function", xlab = "", ylab = "")
points(0:30, density.p, pch = 19, col = "darkred", cex = 0.5)

### QUANTILE FUNCTION:

xx <- seq(0, 1, by = 0.01)
quantile.p <- qpois(xx, lambda = 10)

plot(xx, quantile.p, type = 's', col = "red",
     main = "Quantile function", xlab = "", ylab = "")


# ...................................... B .....................................


# Generate random vector of length 1000 with independent ele-
# ments, where each element X is following a poisson distribution
# Po(lambda = 10). Produce a histogram and compare it to the
# desinty plot of the same distribution.

?rpois

n <- rpois(1000, lambda = 10) # vector of 1000 random numbers
d.p <- dpois(0:30, lambda = 10) # density of binomial distribution

hist(n, ylim = c(0, 0.15), freq = F, col = "red")
points(0:30, d.p, pch = 19, type = "b")
box()


# ----------------------------------- TASK 3 -----------------------------------


# ...................................... A .....................................


# Compute the probability that some realization of X from N(20, 16) lies among
# values 12 and 28 (use the cumulative distribution function).

pnorm(28, mean = 20, sd = sqrt(16)) - pnorm(12, mean = 20, sd = 4)

# F(b) - F(a)


# ...................................... B .....................................


# Find the value of the random variable X from N(20, 16) for which the
# probability that X is lesser or equal to it is exactly 95%.

qnorm(0.95, mean = 20, sd = 4)


# ----------------------------------- TASK 4 -----------------------------------


# ...................................... A .....................................


n <- 50000
p <- 0.5

# 1000 random numbers from the binomial distribution:

X <- rbinom(1000, n, p)

# standardization of the previous vector (from the lecture slaid):

U <- (X - n * p) / sqrt(n * p * (1 - p))

xx <- seq(-3, 3, by = 0.01)
density.n <- dnorm(xx, 0, 1) # density of normal distribution

hist(U, freq = F, col = "red", ylim = c(0, 0.5))
lines(xx, density.n, lwd = 2)

# histogram of the transformed data from the binomial distribution looks quite
# similar to the density function of the normal distribution (thanks to the
# central limit theorem)


# ...................................... B .....................................


# EXTRA TASK

n <- 500
p <- 0.5

x <- dbinom(0:n, n, p)

# probability function of binom. distr. for all possible number of successes 0:n

# use formulas from the lecture for computing
# standart deviation and expactation value of X:

exp.b <- n * p
sd.b <- sqrt(n * p * (1 - p))

# plot the result:

xx <- seq(0, n, by = 0.01)
d.norm <- dnorm(xx, mean = exp.b, sd = sd.b)

plot(0:n, x, xlab = "number of successes", col = "red", pch = 16,
     ylab = "value in Pascal triangle", main = paste("n=", n))
lines(xx, d.norm)

# the probability function of the binomial distribution for large n looks
# similar to the denstiy function of normal distribution, which is the consequence
# of the central limit theorem