SUMMARY
The discussion focuses on computing the expectation \(E(X_{1}^{1/2})\) for a random variable \(X_{1}\) with the density function \(f(x,\theta)=\frac{\exp(-\sqrt{x}/\theta)}{2\theta^2}\). The integral for the expectation is expressed as \(E(X_1^{1/2})=\int_0^{\infty} \sqrt{x}\frac{\exp(-\sqrt{x}/\theta)}{2\theta^2}dx\), which simplifies to \(\int_0^{\infty} t^2\frac{\exp(-t/\theta)}{\theta^2}dt\). The challenge arises from the integration by parts approach, which fails due to the non-vanishing derivative of \(\sqrt{x}\).
PREREQUISITES
- Understanding of probability density functions
- Familiarity with expectation calculations in statistics
- Knowledge of integration techniques, particularly integration by parts
- Basic concepts of exponential functions and their properties
NEXT STEPS
- Study the properties of exponential distributions and their applications
- Learn advanced integration techniques, including substitution and integration by parts
- Explore the concept of moment-generating functions in probability theory
- Investigate the use of numerical integration methods for complex integrals
USEFUL FOR
Statisticians, mathematicians, and students studying probability theory who are interested in expectation calculations and integration techniques.