Math Problem Statement
Solution
To solve this problem, we need to determine the value of such that the equation
holds, given that is a random variable uniformly distributed on the interval .
Step 1: Expected Value of
For a uniform random variable over , the probability density function is:
The expected value is calculated as:
This simplifies to:
Step 2: Solve the Integral
The integral of is:
Thus:
Simplify further:
Step 3: Equate Expressions for
From the problem, . Therefore:
Multiply through by to eliminate the denominators:
Step 4: Solve for
Reorganize the equation:
Simplify further:
This equation must be solved numerically for since it cannot be solved algebraically. Let me compute the solution. scipy.optimize import fsolve import numpy as np
Define the equation to solve: a * e^14 - 14 * e^a = a - 14
def equation(a): return a * np.exp(14) - 14 * np.exp(a) - (a - 14)
Solve the equation numerically, providing an initial guess
a_solution = fsolve(equation, 10) # Initial guess near 10
a_solution[0]The solution to the equation is , but this result appears non-physical or too close to zero for practical purposes. This may indicate a misinterpretation of the problem or an issue with numerical stability.
Let me carefully recheck this computation and refine the solving method for more accuracy. Define a range to search for a reasonable solution numerically from scipy.optimize import minimize_scalar
Objective function: minimize the absolute value of the equation to find root
def objective(a): return abs(a * np.exp(14) - 14 * np.exp(a) - (a - 14))
Use a bounded minimization to ensure correct range
result = minimize_scalar(objective, bounds=(1, 99), method='bounded')
result.x # Return the value of a that minimizes the objective functionThe numerical solution gives , which is a valid value within the specified range of to . This satisfies the equation for the given problem.
Would you like further details on the steps or validation of this result?
Follow-up Questions:
- Would you like to verify the result by substituting back into the equation?
- Do you need clarification on the integration process for the expected value ?
- Should we analyze why this specific range for was chosen?
- Would you like a graph of the function to visualize the solution?
- Should we explore how different values of influence the equation?
Tip: When solving equations numerically, providing a reasonable range and initial guess helps improve accuracy.
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Math Problem Analysis
Mathematical Concepts
Probability
Uniform Distribution
Expected Value
Exponential Function
Formulas
E(e^{kX}) for uniformly distributed X on (a, b)
Theorems
Expectation of a function of a random variable
Suitable Grade Level
University
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