Characteristic Function Integrand Evaluation

In summary: But I'm not sure how to use that information to evaluate the integral.In summary, the characteristic function of the function f(x) = ae^-ax can be determined by evaluating the integral of e^(itx)*ae^-ax from 0 to infinity. This can be done by using the limit of the antiderivative, which is equal to a/(it-a)e^(it-a)x evaluated from 0 to infinity. However, it is important to note that there must be assumptions made about the value of it-a in order to properly evaluate the integral.
  • #1
CivilSigma
227
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Homework Statement


[/B]
I am trying to determien the characteristic function of the function:

$$ f(x)= ae^{-ax}$$

$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$

But I am not sure how to evaluate the integral.

Wolfram alpha suggests this, but I am not sure how to get there.

https://www.wolframalpha.com/input/?i=integral+from+0+to+infitiy+of+e^(itx)*ae^(-ax)dx

The Attempt at a Solution


[/B]
If you just plug in the limits you get (∞ - ...) which is indeterminate.
 
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  • #2
CivilSigma said:

Homework Statement


[/B]
I am trying to determien the characteristic function of the function:

$$ f(x)= ae^{-ax}$$

$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$

But I am not sure how to evaluate the integral.

Wolfram alpha suggests this, but I am not sure how to get there.

https://www.wolframalpha.com/input/?i=integral+from+0+to+infitiy+of+e^(itx)*ae^(-ax)dx

The Attempt at a Solution


[/B]
If you just plug in the limits you get (∞ - ...) which is indeterminate.

You have already evaluated the integral; it is not at all indeterminate. If ##F(t) = \int f(t) \, dt## is the indefinite integral, then ##\int_0^\infty f(t) \, dt = \lim_{t \to \infty} F(t) - F(0).## Remember: you were supposed to have ##a > 0,## but you did not actually say that when you wrote the question.
 
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  • #3
CivilSigma said:
I am trying to determien the characteristic function of the function:
$$ f(x)= ae^{-ax}$$

$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$
The integral is not hard to evaluate if you write it as ##a\int_0^\infty e^{(it -a)x}~dx = \frac a{it - a}e^{(it - a)x}|_0^\infty##. I should add that that last expression should really be ##\lim_{b \to \infty} \frac a{it - a}e^{(it - a)x}|_0^b##. Before going any further, there need to be some assumptions on the value of the expression ##it - a##. You also, can't just "plug in" ##\infty## -- you need to work with a limit as shown above to evaluate the antiderivative.
 
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  • #4
Makes sense. I assumed that it - a < 0 otherwise the integrand diverges to infintiy.
 

Related to Characteristic Function Integrand Evaluation

What is a characteristic function?

A characteristic function is a mathematical function that is used to describe the probability distribution of a random variable. It is defined as the expected value of the exponential function of the random variable.

What is an integrand?

An integrand is the function that is being integrated in a mathematical integral. In the context of characteristic function evaluation, it refers to the function that is being integrated to calculate the characteristic function.

Why is characteristic function integrand evaluation important?

Characteristic function integrand evaluation is important because it allows us to calculate the probability distribution of a random variable, which is essential in many areas of science and statistics. It also allows us to perform various statistical analyses, such as calculating moments and generating random numbers.

What are the methods used for characteristic function integrand evaluation?

There are several methods used for characteristic function integrand evaluation, including numerical integration, series expansion, and asymptotic approximations. The choice of method depends on the complexity of the integrand and the desired level of accuracy.

What are some applications of characteristic function integrand evaluation?

Characteristic function integrand evaluation has many applications in fields such as finance, physics, and engineering. It is used in option pricing, risk management, signal processing, and many other areas where the analysis of random variables is necessary.

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