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I have three random variables X1, X2, X3 which follow independent Poisson distributions with parameters λ1, λ2, λ3 >0 and then the random variables X = X1 + X3 and Y = X2 + X3 jointly follow a bivariate Poisson distribution BP(λ1, λ2, λ3). If I only know $$ P(X > Y) = constant number 1\\ P(X = Y) = constant number 2 \\ P(X < Y) = constant number 3 $$, how can I get λ1, λ2, λ3?

If it's impossible to calculate, assuming X and Y are in this range [0.001, 4], can we still get the approximated values?

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  • $\begingroup$ Can you write those probabilities as functions of the lambdas? $\endgroup$
    – Glen_b
    Aug 19 at 23:59
  • $\begingroup$ I've just edited my question. If I only have those probabilities as constant numbers, can I calculate lambdas? $\endgroup$
    – Juan
    Aug 20 at 1:15
  • $\begingroup$ Naturally you would have the probability values as numbers, that's what probabilities are. To repeat: Can you write those probabilities as functions of the lambdas?. ... (... so, of course, you can use the numbers as the right hand sides of 3 equations in 3 unknowns, to hopefully solve for the three lambda parameters) $\endgroup$
    – Glen_b
    Aug 20 at 5:54
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    $\begingroup$ I would expect this to be unsolvable generally, because your three constants must sum to unity. Moreover, all three can be expressed in terms of $X-Y=X_1-X_2,$ showing it's impossible to find $\lambda_3:$ its value could be anything. $\endgroup$
    – whuber ♦
    Aug 20 at 14:43
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    $\begingroup$ Indeed. The answer to my question to the OP (which I assumed they'd figure out once they proceeded to attempt to manipulate the equations) is that you don't end up with three equations in three unknowns; because each of the known probabilities involves $X-Y$ there's really only information about two of them at best. $\endgroup$
    – Glen_b
    Aug 21 at 0:17

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