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

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The discussion centers on determining the distribution of a random variable x defined by the equation x = a + b*z + c*y, where z follows a normal distribution and y follows an exponential distribution. It highlights the independence of the two random variables and focuses on the convolution of their distributions to find the distribution of x. The process involves integrating the probability densities of z and y to derive the overall distribution of x. Additionally, it notes the constraints on z and y based on their respective distributions, particularly regarding the non-negativity of y. The conversation also touches on the broader topic of error propagation and the (co)variance of functions of random variables.
Steve Zissou
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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!
 
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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 ##
 
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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.
 
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