- #1

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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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- Thread starter Steve Zissou
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- #1

- 36

- 0

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!

- #2

Science Advisor

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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 ##

- #3

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Nice! Thank you very much, FactChecker!

- #4

Science Advisor

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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.)

- #5

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Thanks, mjc123!

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