independent identically distributed random variables
For two independent and identically distributed random variables having the exponential distribution, do they have the same lambda value, or are the lambda values different?

If the have the same distribution function, then they both must use the same lambda.

I just wasn't sure b/c it looks like Poisson Random Variables can have different lambda parameters.

That is true.

Yes, they can, but your original post said "identically distributed". That doesn't mean only that the both have an exponential distribution but that they have the same exponential distribution. If they are "identically distributed" then everything about the distributions is the same.

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