We are given the additional information that conditional on the parameter of interest \lambda, our observations X_1, X_2, \ldots, X_ n are independently and identically distributed according to the probability distribution \textsf{Poiss}(\lambda ). From here, we compute our likelihood function. Our approach would be to compute a single function of \lambda L(X_ i | \lambda ), then plug in our data X_1, X_2, \ldots, X_ n to compute the overall likelihood L_ n(X_1, X_2, \ldots ,X_ n | \lambda ) = L(X_1 | \lambda )L(X_2|\lambda )\ldots L(X_ n|\lambda ).
In our framework so far, we treat the observations X_1, \ldots , X_ n as fixed values, by which we perform Bayesian inference. Hence, L(X_ i | \lambda ) is to be viewed as a function of \lambda with X_ i as a parameter. Thus when using proportionality notation, we only need to consider variations concerning our parameter of interest \lambda. Compute the likelihood function L(X_1|\lambda ), using proportionality notation to simplify it such that in your expression, L(X_1|\lambda =1)=e^{-1} regardless of the value of X_1. (Note that this is not necessarily the actual likelihood L.)
Use X1 for X_1.
L(X_1|\lambda ) \propto
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Multiply the expressions L(X_1|\lambda ), L(X_2|\lambda ), \ldots, L(X_ n|\lambda ) based on the simplified expression for L(X_1|\lambda ) to get the desired likelihood function L_ n(X_1, X_2, \ldots ,X_ n | \lambda ).
Use SumXi for \sum _{i=1}^ n X_ i.
L_ n(X_1, X_2, \ldots ,X_ n | \lambda ) \propto
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