# How to solve an expression with inverse of exponential distribution

• MHB
• user_01
In summary, one possible way to solve the problem of infinite expected value of E in energy harvesting expression is to use a truncated exponential distribution with an upper bound on the random variable h, which will give a finite and more practical expected value for E.
user_01
I have an Energy harvesting expression something like the following
$R = \tau B \log\Big(1 + \frac{E h^2}{\tau r^\alpha\sigma^2} \Big)$
$E = \tau(2^{R/\tau B}-1 )\frac{r^\alpha\sigma^2}{h^2}$

Let all constant terms as $a$ to simplify the expression into : $E = a\frac{1}{h^2}$
$E$ is a random variable because of $h^2\sim\exp(1)$ i.e. $h$ is exponentially distributed random variable with unit parameter.

Consider $x= h^2 \sim \exp(1)$,
$\mathbb{E}[E] = E_x= a \mathbb{E}\Big[\frac{1}{x}\Big] = a\int_0^\infty \frac{e^{-x}}{x} \to \infty\ (\text{Divergent})$

I have to solve this problem because my circuit cannot harvest an infinite amount of energy (this is impractical).
How can I solve this problem? Any way around to find the approximate solution?

I will really appreciate any suggestions.

One possible solution is to use a truncated exponential distribution. This means that you can bound the range of possible values for h and use the truncated distribution to compute the expected value of E.For example, if you set a hard upper bound of h at h_max, then you can compute the expected value of E as: $E_{x|h<h_{max}} = a \mathbb{E}\Big[\frac{1}{x}\Big]_{x<h_{max}^2} = a\int_0^{h_{max}^2} \frac{e^{-x}}{x}$This should give you a finite expected value of E that is much better than the original divergent result.

## 1. How do I find the inverse of an exponential distribution?

To find the inverse of an exponential distribution, you can use the formula x = ln(1-y)/(-λ), where x is the inverse of the distribution, y is a random number between 0 and 1, and λ is the rate parameter of the distribution.

## 2. What is the purpose of using the inverse of an exponential distribution?

The inverse of an exponential distribution is often used to solve for a specific value of x, given a certain probability. This is useful in many applications, such as in finance and engineering, where knowing the probability of a certain outcome is important.

## 3. Can the inverse of an exponential distribution be negative?

Yes, the inverse of an exponential distribution can be negative. This can occur when the value of y in the formula is close to 1, resulting in a negative value for x. However, in most cases, the inverse will be a positive value.

## 4. What is the relationship between the inverse of an exponential distribution and the exponential function?

The inverse of an exponential distribution is closely related to the exponential function. The exponential function is used to model the probability of a certain event occurring over a continuous period of time, while the inverse of the exponential distribution is used to solve for a specific value of x given a certain probability.

## 5. Are there any limitations to using the inverse of an exponential distribution?

One limitation of using the inverse of an exponential distribution is that it assumes a constant rate of decay. In real-world situations, this may not always be the case. Additionally, the inverse of an exponential distribution may not accurately model data that is heavily skewed or has outliers.

Replies
1
Views
4K
Replies
1
Views
995
Replies
2
Views
832
Replies
1
Views
1K
Replies
1
Views
2K
Replies
25
Views
2K
Replies
8
Views
1K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K