input: def predict_LTV(customer_input: np.ndarray) -> float:
      """Predict the customer lifetime value (LTV) given a customer's past transactions.
      The function reads past transaction data from 'transactions.csv', preprocesses it,
      and trains a RandomForestRegressor model. The function then takes a new customer
      input (a 2D numpy array of past transactions) and returns the predicted LTV.
      
      >>> predict_LTV(np.array([[10, 100, 5, ...]]))
      503.25
      """
output:     import numpy as np
      import pandas as pd
      from sklearn.ensemble import RandomForestRegressor
      from sklearn.model_selection import train_test_split
      from sklearn.preprocessing import StandardScaler
  
      # Load the transaction data
      df = pd.read_csv('transactions.csv')
  
      # Preprocess the data
      X = df.drop('LTV', axis=1).values
      y = df['LTV'].values
  
      # Split the data into training and testing sets
      X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
  
      # Standardize the features
      scaler = StandardScaler()
      X_train = scaler.fit_transform(X_train)
      X_test = scaler.transform(X_test)
  
      # Train the RandomForestRegressor model
      model = RandomForestRegressor(n_estimators=100, random_state=42)
      model.fit(X_train, y_train)
  
      # Preprocess the customer input
      customer_input = scaler.transform(customer_input)
  
      # Predict the LTV for the customer input
      predicted_ltv = model.predict(customer_input)
  
      return predicted_ltv[0]