Integral involving exponential logarithm

In summary, the expected value of X, or E[X], is equal to 1 over lambda. To find this, you can use integration and let u = lambda x, which simplifies the integral to 1 over lambda times the integral of ue^-u du. However, it is important to remember to replace the "x" in the original function with u/lambda.
  • #1
cielo
15
0

Homework Statement


If X is a random variable with density function: f(x) = [tex]\lambda[/tex]e[tex]^{-x \lambda}[/tex]where X>=0.

Homework Equations


Why is the expected value of X, or E[X] = [tex]\frac{1}{\lambda}[/tex]?

The Attempt at a Solution


E[X] = [tex]\int[/tex] x*([tex]\lambda[/tex]e[tex]^{- \lambda}[/tex][tex]^{x}[/tex]) dx, where the integral is from 0 to infinity.

I let u = -[tex]\lambda[/tex]x
du = -[tex]\lambda[/tex] dx

but I can't get the [tex]\frac{1}{\lambda}[/tex] as the answer when I performed the integration.
 
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  • #2
If you let [itex]u= \lambda x[/itex] then [itex]x= u/\lambda[/itex] and the integral becomes
[tex]\int \frac{u}{\lambda}e^{-u} du= \frac{1}{\lambda}\int ue^{-u}du[/tex]

Did you remember to replace the "x" multiplying the exponential with [itex]u/\lambda[/itex]?
 

FAQ: Integral involving exponential logarithm

What is the general form of an integral involving exponential logarithm?

The general form of an integral involving exponential logarithm is ∫(logax)n * ex dx, where n is any real number and a is the base of the logarithm.

How do you solve an integral involving exponential logarithm?

To solve an integral involving exponential logarithm, you can use integration by parts or substitution. You can also use properties of logarithms and exponentials to simplify the integral before integrating.

Can an integral involving exponential logarithm be evaluated using a closed form solution?

Yes, some integrals involving exponential logarithm can be evaluated using a closed form solution. However, for more complex integrals, a numerical approximation may be necessary.

What is the significance of integrals involving exponential logarithm in mathematics?

Integrals involving exponential logarithm are important in various areas of mathematics, including calculus, differential equations, and complex analysis. They also have applications in physics, engineering, and other fields of science.

Are there any special techniques for solving integrals involving exponential logarithm?

Yes, there are some special techniques for solving integrals involving exponential logarithm, such as using trigonometric substitutions or converting the integral into a series representation. It is also helpful to have a good understanding of properties of logarithms and exponentials.

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