# Distribution of Sum of Two Weird Random Variables....

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• Steve Zissou
In summary, the conversation discusses the relationship between different distributions and how to determine the distribution of a sum of two random variables. The method of convolution is used to calculate the distribution of the sum, and the process is demonstrated with an example. The topic of Propagation of Error/Uncertainty is also mentioned as a way to determine the covariance of functions of random variables.
Steve Zissou
Hi there.

Let's say I have the following relationship:

x = a + b*z + c*y

z is distributed normally
y is distributed according to a different distribution, say exponential

Is there a way to figure out what is the distribution of x?

Thanks!

I will assume that the two random variables are independent.
The constant offset, ##a##, can always be dealt with last. So let's ignore it for now.
Consider the sum of two random variables, ##r_1 = b*z## and ##r_2 = c*y## with distributions ##p_1(r_1)## and ##p_2(r_2)##, respectively.
The distribution, ##p(x)##, of the sum, ##x = r_1+r_2## is the convolution, ##p(x) = \int_{t=-\infty}^{t=\infty}p_1(t)p_2(x-t) \,dt ##

Steve Zissou
Nice! Thank you very much, FactChecker!

For a given value x = X,
What is the probability that z has a value Z?
What is the probability that y has the value Y = (X -a -bZ)/c?
Integrate over z: f(x=X) = ∫f(z=Z)f(Y=(X-a-bZ)/c)dZ
(Assuming z and y are distributed independently. If not, you have to use a conditional probability for Y.)
Note that the range of z and y may be limited to less than their full possible range, e.g. if z is normal, it can take negative values, but if y is exponential it can only be positive (or zero). Therefore Z is limited to values for which X - a - bZ is nonnegative, ie Z ≤ (X - a)/b. (That's if b and c are both positive, work it out for yourself for other cases.)

Thanks, mjc123!

Thought I would generalize by referring to the topic of Propagation of Error/Uncertainty, who's topic is to determine the (co)variance of functions of Random Variables.

## What is meant by "weird" random variables in this context?

In the context of the distribution of the sum of two weird random variables, "weird" typically refers to random variables that do not follow conventional or well-known probability distributions such as the normal or uniform distributions. These could be variables with highly skewed, multimodal, or heavy-tailed distributions, or those that follow a rule or pattern that is not commonly encountered in statistical practice.

## How do you find the distribution of the sum of two weird random variables?

To find the distribution of the sum of two weird random variables, one generally needs to use convolution if the variables are independent. Convolution involves integrating the product of the two individual distributions over all possible values, which can be quite complex depending on the form of the distributions. For discrete random variables, the sum's distribution is found by summing the products of probabilities for each pair of outcomes that result in each possible sum.

## Are there any special techniques or considerations when dealing with weird random variables?

Yes, special techniques may be required when dealing with weird random variables. These might include using numerical methods or simulations such as Monte Carlo methods, particularly when analytical solutions are difficult or impossible to obtain. It's also crucial to carefully analyze the properties of the distributions, such as their tails and moments, to understand how they might affect the sum or other aggregate statistics.

## What are some examples of applications where the sum of weird random variables is relevant?

Applications of the sum of weird random variables can be found in various fields such as finance, where the returns on certain types of investments might not follow standard distributions; in insurance, for modeling losses from rare but severe events; or in physics and engineering, where certain phenomena may be modeled more accurately with non-standard distributions. Understanding the sum of these variables helps in risk assessment, pricing, and other decision-making processes.

## What challenges might one face when working with the sum of weird random variables?

Challenges in working with the sum of weird random variables include difficulties in deriving exact distributional forms, computational complexities, and the potential for increased error or uncertainty in estimation and modeling. Additionally, the lack of familiarity and intuitive understanding of these distributions can make it harder to predict their behavior and implications in practical scenarios.

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