code: 
  '''simple docstring'''
  
  
  
  
  
  from collections.abc import Callable
  from math import log
  from os import name, system
  from random import gauss, seed
  from typing import TypeVar
  def   snake_case_       (							_a     :							float ,					_a     :							float ,					_a     :							int			):
        seed(1			)
        return [gauss(_a ,					_a			) for _ in range(_a			)]
  def   snake_case_       (							_a     :							int ,					_a     :							list			):
        return [k for k in range(_a			) for _ in range(instance_count[k]			)]
  def   snake_case_       (							_a     :							int ,					_a     :							list			):
        return sum(_a			) / instance_count
  def   snake_case_       (							_a     :							int ,					_a     :							int			):
        return instance_count / total_count
  def   snake_case_       (							_a     :							list ,					_a     :							list ,					_a     :							int			):
        snake_case_						       =							[]  # An empty list to store all squared differences
        # iterate over number of elements in items
        for i in range(len(_a			)			):
              # for loop iterates over number of elements in inner layer of items
              for j in range(len(items[i]			)			):
                    # appending squared differences to 'squared_diff' list
                    squared_diff.append((items[i][j] - means[i]) ** 2			)
  
      # one divided by (the number of all instances - number of classes) multiplied by
      # sum of all squared differences
        snake_case_						       =							len(_a			)  # Number of classes in dataset
        return 1 / (total_count - n_classes) * sum(_a			)
  def   snake_case_       (							_a     :							list ,					_a     :							list ,					_a     :							float ,					_a     :							list			):
        snake_case_						       =							[]
        # for loop iterates over number of elements in list
        for i in range(len(_a			)			):
              # for loop iterates over number of inner items of each element
              for j in range(len(x_items[i]			)			):
                    snake_case_						       =							[]  # to store all discriminant values of each item as a list
                    # for loop iterates over number of classes we have in our dataset
                    for k in range(len(_a			)			):
                          # appending values of discriminants for each class to 'temp' list
                          temp.append(
                              x_items[i][j] * (means[k] / variance)
                              - (means[k] ** 2 / (2 * variance))
                              + log(probabilities[k]			)			)
                    # appending discriminant values of each item to 'results' list
                    results.append(_a			)
  
        return [result.index(max(_a			)			) for result in results]
  def   snake_case_       (							_a     :							list ,					_a     :							list			):
        snake_case_						       =							sum(1 for i, j in zip(_a ,					_a			) if i == j			)
        # percentage of accuracy equals to number of correct predictions divided by number
        # of all data and multiplied by 100
        return (correct / len(_a			)) * 100
  
  
  A           =   TypeVar("num")
  def   snake_case_       (							_a     :							Callable[[object], num] ,					_a     :							str ,					_a     :							str ,					_a     :							Callable[[num], bool] = lambda _a			: True ,					_a     :							str | None = None ,					):
        while True:
              try:
                    snake_case_						       =							input_type(input(_a			).strip() or default			)
                    if condition(_a			):
                          return user_input
                    else:
                          print(F'''{user_input}: {err_msg}'''			)
                          continue
              except ValueError:
                    print(
                        F'''{user_input}: Incorrect input type, expected {input_type.__name__!r}'''			)
  def   snake_case_       (							):
        while True:
              print(" Linear Discriminant Analysis ".center(50 ,					"*"			)			)
              print("*" * 50 ,					"\n"			)
              print("First of all we should specify the number of classes that"			)
              print("we want to generate as training dataset"			)
              # Trying to get number of classes
              snake_case_						       =							valid_input(
                  input_type=_a ,					condition=lambda _a			: x > 0 ,					input_msg="Enter the number of classes (Data Groupings): " ,					err_msg="Number of classes should be positive!" ,					)
  
              print("-" * 100			)
  
              # Trying to get the value of standard deviation
              snake_case_						       =							valid_input(
                  input_type=_a ,					condition=lambda _a			: x >= 0 ,					input_msg=(
                      "Enter the value of standard deviation"
                      "(Default value is 1.0 for all classes): "
                  ) ,					err_msg="Standard deviation should not be negative!" ,					default="1.0" ,					)
  
              print("-" * 100			)
  
              # Trying to get number of instances in classes and theirs means to generate
              # dataset
              snake_case_						       =							[]  # An empty list to store instance counts of classes in dataset
              for i in range(_a			):
                    snake_case_						       =							valid_input(
                        input_type=_a ,					condition=lambda _a			: x > 0 ,					input_msg=(F'''Enter The number of instances for class_{i+1}: ''') ,					err_msg="Number of instances should be positive!" ,					)
                    counts.append(_a			)
              print("-" * 100			)
  
              # An empty list to store values of user-entered means of classes
              snake_case_						       =							[]
              for a in range(_a			):
                    snake_case_						       =							valid_input(
                        input_type=_a ,					input_msg=(F'''Enter the value of mean for class_{a+1}: ''') ,					err_msg="This is an invalid value." ,					)
                    user_means.append(_a			)
              print("-" * 100			)
  
              print("Standard deviation: " ,					_a			)
              # print out the number of instances in classes in separated line
              for i, count in enumerate(_a ,					1			):
                    print(F'''Number of instances in class_{i} is: {count}'''			)
              print("-" * 100			)
  
              # print out mean values of classes separated line
              for i, user_mean in enumerate(_a ,					1			):
                    print(F'''Mean of class_{i} is: {user_mean}'''			)
              print("-" * 100			)
  
              # Generating training dataset drawn from gaussian distribution
              snake_case_						       =							[
                  gaussian_distribution(user_means[j] ,					_a ,					counts[j]			)
                  for j in range(_a			)
              ]
              print("Generated Normal Distribution: \n" ,					_a			)
              print("-" * 100			)
  
              # Generating Ys to detecting corresponding classes
              snake_case_						       =							y_generator(_a ,					_a			)
              print("Generated Corresponding Ys: \n" ,					_a			)
              print("-" * 100			)
  
              # Calculating the value of actual mean for each class
              snake_case_						       =							[calculate_mean(counts[k] ,					x[k]			) for k in range(_a			)]
              # for loop iterates over number of elements in 'actual_means' list and print
              # out them in separated line
              for i, actual_mean in enumerate(_a ,					1			):
                    print(F'''Actual(Real) mean of class_{i} is: {actual_mean}'''			)
              print("-" * 100			)
  
              # Calculating the value of probabilities for each class
              snake_case_						       =							[
                  calculate_probabilities(counts[i] ,					sum(_a			)			) for i in range(_a			)
              ]
  
              # for loop iterates over number of elements in 'probabilities' list and print
              # out them in separated line
              for i, probability in enumerate(_a ,					1			):
                    print(F'''Probability of class_{i} is: {probability}'''			)
              print("-" * 100			)
  
              # Calculating the values of variance for each class
              snake_case_						       =							calculate_variance(_a ,					_a ,					sum(_a			)			)
              print("Variance: " ,					_a			)
              print("-" * 100			)
  
              # Predicting Y values
              # storing predicted Y values in 'pre_indexes' variable
              snake_case_						       =							predict_y_values(_a ,					_a ,					_a ,					_a			)
              print("-" * 100			)
  
              # Calculating Accuracy of the model
              print(F'''Accuracy: {accuracy(_a ,					_a			)}'''			)
              print("-" * 100			)
              print(" DONE ".center(100 ,					"+"			)			)
  
              if input("Press any key to restart or 'q' for quit: "			).strip().lower() == "q":
                    print("\n" + "GoodBye!".center(100 ,					"-"			) + "\n"			)
                    break
              system("cls" if name == "nt" else "clear"			)  # noqa: S605
  
  
  if __name__ == "__main__":
       main()
  
  
style_context: 
  '''simple docstring'''
  
  
  
  
  
  from __future__ import annotations
  
  A           =   [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]
  
  
  
  class  lowerCAmelCase__   :
  
  
  
    '''simple docstring'''
  
  
  
  
  
  
    def __init__(self  : Optional[int]	, lowerCAmelCase_  : int    )   ->       None:
          snake_case_						       =							-1
          snake_case_						       =							-1
          snake_case_						       =							[-1] * length
          snake_case_						       =							None
          snake_case_						       =							None
  
  
    def __repr__(self  : List[Any]    )   ->       str:
          return f'''Node(min_value={self.minn} max_value={self.maxx})'''
  
  def   snake_case_       (							_a     :							list[int]			):
        snake_case_						       =							Node(len(_a			)			)
        snake_case_       ,	snake_case_						       =							min(_a			), max(_a			)
        # Leaf node case where the node contains only one unique value
        if root.minn == root.maxx:
              return root
        snake_case_						       =							(root.minn + root.maxx) // 2
  
        snake_case_						       =							[]
        snake_case_						       =							[]
  
        for index, num in enumerate(_a			):
              if num <= pivot:
                    left_arr.append(_a			)
              else:
                    right_arr.append(_a			)
              snake_case_						       =							len(_a			)
        snake_case_						       =							build_tree(_a			)
        snake_case_						       =							build_tree(_a			)
        return root
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int			):
        if index < 0 or node is None:
              return 0
        # Leaf node cases
        if node.minn == node.maxx:
              return index + 1 if node.minn == num else 0
        snake_case_						       =							(node.minn + node.maxx) // 2
        if num <= pivot:
              # go the left subtree and map index to the left subtree
              return rank_till_index(node.left ,					_a ,					node.map_left[index] - 1			)
        else:
              # go to the right subtree and map index to the right subtree
              return rank_till_index(node.right ,					_a ,					index - node.map_left[index]			)
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int ,					_a     :							int			):
        if start > end:
              return 0
        snake_case_						       =							rank_till_index(_a ,					_a ,					_a			)
        snake_case_						       =							rank_till_index(_a ,					_a ,					start - 1			)
        return rank_till_end - rank_before_start
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int ,					_a     :							int			):
        if index > (end - start) or start > end or node is None:
              return -1
        # Leaf node case
        if node.minn == node.maxx:
              return node.minn
        # Number of elements in the left subtree in interval [start, end]
        snake_case_						       =							node.map_left[end] - (
            node.map_left[start - 1] if start else 0
        )
        if num_elements_in_left_tree > index:
              return quantile(
                  node.left ,					_a ,					(node.map_left[start - 1] if start else 0) ,					node.map_left[end] - 1 ,					)
        else:
              return quantile(
                  node.right ,					index - num_elements_in_left_tree ,					start - (node.map_left[start - 1] if start else 0) ,					end - node.map_left[end] ,					)
  def   snake_case_       (							_a     :							Node | None ,					_a     :							int ,					_a     :							int ,					_a     :							int ,					_a     :							int			):
        if (
            start > end
            or node is None
            or start_num > end_num
            or node.minn > end_num
            or node.maxx < start_num
        ):
              return 0
        if start_num <= node.minn and node.maxx <= end_num:
              return end - start + 1
        snake_case_						       =							range_counting(
            node.left ,					(node.map_left[start - 1] if start else 0) ,					node.map_left[end] - 1 ,					_a ,					_a ,					)
        snake_case_						       =							range_counting(
            node.right ,					start - (node.map_left[start - 1] if start else 0) ,					end - node.map_left[end] ,					_a ,					_a ,					)
        return left + right
  
  
  if __name__ == "__main__":
       import doctest
  
       doctest.testmod()
  
  