# Mathematica unable to realize simple distributional relationship

I was trying to find the CDF for the following problem:

In:= Probability[e1 - Log[e2 + e3] <= t, {
e1 \[Distributed] ExponentialDistribution,
e2 \[Distributed] ExponentialDistribution,
e3 \[Distributed] ExponentialDistribution}]

Out= Probability[
e1 - Log[e2 + e3] <=
t, {e1 \[Distributed] ExponentialDistribution,
e2 \[Distributed] ExponentialDistribution,
e3 \[Distributed] ExponentialDistribution}]


Clearly Mathematica didn't like that. Then I rephrased it as

In:= Probability[e1 - Log[c] <= t, {
e1 \[Distributed] ExponentialDistribution,

Out= E^-E^-t


which apparently wasn't a problem Note that the sum of two standard exponentials is a gamma distribution with parameters (2,1). So the two problems should have the same solution.

Is there some way I can help Mathematica realize this? Maybe I need to specify something about the numbers not being complex or something?

(Note, I get the same results whether I'm using Mathematica 12.1.1.0 on my own machine, or Wolfram Cloud.)

Update: If I switch the order of the arguments, it also stops working with the GammaDistribution:

In:= Probability[
e1 - Log[c] <= t, {c \[Distributed] GammaDistribution[2, 1],
e1 \[Distributed] ExponentialDistribution}]
Out= Probability[
e1 - Log[c] <= t, {c \[Distributed] GammaDistribution[2, 1],
e1 \[Distributed] ExponentialDistribution}]


It's definitely very brittle.

• Funny CDF[TransformedDistribution[ e1 - Log[e2 + e3], {e1 \[Distributed] ExponentialDistribution, e2 \[Distributed] ExponentialDistribution, e3 \[Distributed] ExponentialDistribution}], t] works. Mar 15 at 8:45
• If you add Assumptions -> t > 0, it works. (Windows 10 12.3.1.0 and 13.2.0)
– JimB
Mar 15 at 13:17
• Thanks! Both of these work! But JimB's solution is faster :) Mar 15 at 19:36
• Is t being postive required? Mar 15 at 20:44
• @ВалерийЗаподовников good point. The CDF is valid for all $t$, also negative. Mar 16 at 20:52