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Suppose $X_t\sim Poisson(\lambda t)$. I'm looking at an exercise problem where two probability statements are given, and the exercise is to algebraically determine the value of a third probability:

$P(X_1=1)=0.36788$

$P(X_2=3)=0.18405$

$P(X_1=2)=\ ?$

I know $\lambda$ could easily be solved for algebraically if $P(X_t=0)$ is given for any $t$... but in this case, the PMF of $X_t$ can't be used to algebraically solve for $\lambda$.

The exercise says to use an algebraic approach -- not to manually try to find $\lambda$ through guesswork or Newton Raphson. So I want to ignore the fact that I could simply inspect and see that $\lambda=1$ and use that to solve the requested probability. What is the algebraic way to approach this, that would work in cases where $\lambda$ wasn't a simple value such as 1?

Jonathan
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    Because $\Pr(X_1=3)/\Pr(X_1=1)=\lambda^2/6,$ you can solve this algebraically for $\lambda.$ You will discover the solution is not a "simple value," but it is easily found. – whuber Jul 22 '22 at 16:14
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    Jonathan does not know $P(X_1=3)$. But of course, your idea is right. – FP0 Jul 22 '22 at 20:24
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    @FPO Good catch--I overlooked the subscript "2" in the second equation. – whuber Jul 22 '22 at 21:04

1 Answers1

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Since $X_t\sim\text{Pois}(\lambda t)$, $$\mathbb{P}(X_t=k)=e^{-\lambda t}\frac{\left(\lambda t\right)^k}{k!}$$ As a consequence, you can develop expressions of your values $\mathbb{P}(X_1=1)$ and $\mathbb{P}(X_2=3)$ as functions of $\lambda$. Then, you can find a couple of ways to use these 2 formulas and the numerical values you know in order to determine the value of $\lambda$.

Hint:

Determine the formula for $\mathbb{P}(X_1=1)^2$.

FP0
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