Difficult integral (product of gamma func with exp)

In summary: Therefore, the final solution is:In summary, the solution to the given integral is -\frac{1}{2n}\ln\left|2n\theta\right|.
  • #1
coolnessitself
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Homework Statement


Solve. [tex]\int\limits_{0}^\infty \exp\left\{\frac{-2n}{x} - \frac{x}{\theta}\right\} x^{n-1} dx[/tex]

[tex]n,\theta[/tex] are constants.

Homework Equations



The Attempt at a Solution


So actually this problem is from the realm of stats...find the MVUE of [tex]P(X<2)[/tex] where [tex]X_i\sim \mathrm{Exp}(\theta)[/tex]. The MLE of an exp is [tex]\overline{X}[/tex], and this is invariant under transforms, and so we need to solve the expectation of [tex]1-\exp\left\{\frac{-2n}{\overline{x}}\right\}[/tex]. So the 1 is easy, and since here [tex]\overline{X}\sim\Gamma[/tex], it's the expectation of [tex]\frac{-2n}{x}\right\}[/tex] in a gamma distribution, resulting in the integral above, with some constants I've removed. The exam is over, no one could find a solution. Can the integral be solved? I tried a u sub for sqrt(y..) and tried intrgrating by parts n-1 times, and a few other ideas. Any ideas for a method to solve the above integral?
 
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  • #2


First of all, great job on recognizing that this problem is related to statistics and the MLE of an exponential distribution. To solve this integral, we can use the substitution u = 1/x, which gives us du = -1/x^2 dx. We can then rewrite the integral as:

\int\limits_{0}^\infty \exp\left\{\frac{-2n}{x} - \frac{x}{\theta}\right\} x^{n-1} dx = \int\limits_{0}^\infty \exp\left\{-2nu - \frac{1}{\theta u}\right\} u^{n+1} du

We can then use the fact that the integral of an exponential function is equal to its reciprocal to rewrite the integral as:

\int\limits_{0}^\infty \frac{u^{n+1}}{\exp\left\{2nu + \frac{1}{\theta u}\right\}} du

Next, we can use the substitution v = 2nu + 1/(theta u), which gives us dv = 2n - 1/(theta u^2) du. We can then rewrite the integral as:

\int\limits_{0}^\infty \frac{u^{n+1}}{\exp\left\{v\right\}} du = \int\limits_{0}^\infty \frac{1}{2n - 1/(theta u^2)} du

Using partial fractions, we can rewrite this integral as:

\int\limits_{0}^\infty \frac{1}{2n}\left(\frac{1}{u - \frac{1}{2n\theta}} + \frac{1}{u + \frac{1}{2n\theta}}\right) du

Integrating each term separately, we get:

\frac{1}{2n}\left(\ln\left|u - \frac{1}{2n\theta}\right| - \ln\left|u + \frac{1}{2n\theta}\right|\right)\Bigg|_0^\infty

Since the limits of integration are from 0 to infinity, the first term evaluates to 0. The second term evaluates to:

\frac{1}{2n}\left(\ln\left|2n\theta\right|
 

Related to Difficult integral (product of gamma func with exp)

1. What is a difficult integral?

A difficult integral is a type of mathematical problem that involves finding the area under a curve or the volume of a shape, but with a particularly complex or challenging function to integrate.

2. How do you solve a difficult integral?

Solving a difficult integral typically involves using advanced techniques such as substitution, integration by parts, or trigonometric identities. It may also require knowledge of special functions, series expansions, or numerical methods.

3. What is the product of gamma function with exponential?

The product of the gamma function with an exponential function is a type of integral that involves multiplying the gamma function, which is a generalization of the factorial function, with an exponential function, which is a function of the form e^x.

4. Why is the integral of the product of gamma function with exp difficult to solve?

The integral of the product of the gamma function with an exponential function is difficult to solve because it involves two complex functions and requires advanced integration techniques. Additionally, the gamma function does not have a simple closed form solution, making the integral even more challenging.

5. What are some real-world applications of difficult integrals?

Difficult integrals have numerous applications in physics, engineering, and other scientific fields. They are often used to calculate quantities such as electric fields, gravitational potentials, and probabilities in quantum mechanics. They are also used in statistical analysis, signal processing, and other areas of mathematics.

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