Code

 import numpy as np

from scipy.stats import skew

x = np.array([4, 5, 6, 7, 8, 9, 10])

f = np.array([2, 3, 2, 5, 3, 4, 2])

data = np.repeat(x, f)

skewness = skew(data)

print(f"Skewness: {skewness:.10f}")

mean = np.mean(data)

mu_4 = np.mean((data - mean)**4)

print(f"Fourth central moment:

 {mu_4:.10f}")

import numpy as np

from scipy.stats import kurtosis

x = np.array([1, 2, 3, 4, 5])

f = np.array([2, 3, 4, 5, 6])

data = np.repeat(x, f)

kurt = kurtosis(data)

print(f"Kurtosis: {kurt:.10f}")

mean = np.mean(data)

mu_4 = np.mean((data - mean)**4)

print(f"Fourth central moment:

 {mu_4:.10f}")

from math import comb

n = 6

p = 0.1

def binomial_prob(n, k, p):

    return comb(n, k) * (p ** k) * ((1 - p) ** (n - k))

p_r_success = binomial_prob(6, 1, p)

p_at_least_r = binomial_prob(6, 1, p) + binomial_prob(6, 2, p) + binomial_prob(6, 3, p)

p_at_most_r = 1 - binomial_prob(6, 1, p) - binomial_prob(6, 2, p)

print("P(X = r):", p_r_success)

print("P(X >= r):", p_at_least_r)

print("P(X <= r)

:", p_at_most_r)

import numpy as np

from scipy.stats import poisson

# Given data

accidents = [0, 1, 2, 3, 4]

frequency = [18, 27, 25, 10, 5]

# Total shifts and total accidents

total_shifts = sum(frequency)

total_accidents = sum([x * p for x, p in zip(accidents, frequency)])

# Calculate mean (λ)

mean_lambda = total_accidents / total_shifts

# Calculate expected frequency using Poisson distribution

expected_frequency = [total_shifts * poisson.pmf(k, mean_lambda) for k in accidents]

# Output

print("Fitting Poisson distribution:")

print("Mean (λ) =", mean_lambda)

print("Expected Frequencies =",

 expected_frequency)