[mathjaxinline]\pi (\lambda )L_ n(X_1, X_2, \cdots , X_ n|\lambda )[/mathjaxinline].
The likelihood function [mathjaxinline]L_n(X_1, X_2, \cdots , X_n | \lambda)[/mathjaxinline] is the probability of observing the data [mathjaxinline]X_1, X_2, \cdots, X_n[/mathjaxinline] given a value of [mathjaxinline]\lambda[/mathjaxinline].
To compute [mathjaxinline]\pi (\lambda | X_1, X_2, \cdots , X_ n)[/mathjaxinline], we need to multiply this likelihood function by the prior distribution [mathjaxinline]\pi(\lambda)[/mathjaxinline].
Let's assume that the prior distribution [mathjaxinline]\pi(\lambda)[/mathjaxinline] is a density function that represents our prior beliefs or knowledge about the value of [mathjaxinline]\lambda[/mathjaxinline].
Therefore, [mathjaxinline]\pi(\lambda | X_1, X_2, \cdots , X_n) \propto \pi(\lambda) L_n(X_1, X_2, \cdots , X_n | \lambda)[/mathjaxinline]. This expression represents the posterior distribution of [mathjaxinline]\lambda[/mathjaxinline] given the observed data [mathjaxinline]X_1, X_2, \cdots, X_n[/mathjaxinline], up to a constant of proportionality.
According to Bayes' formula, [mathjaxinline]\pi (\lambda | X_1, X_2, \cdots , X_ n) \propto \pi (\lambda )L_ n(X_1, X_2, \cdots , X_ n|\lambda )[/mathjaxinline]. This yields the posterior distribution up to a constant of proportionality. Multiply the relevant expressions above (use the simplified version with proportionality notation) to compute [mathjaxinline]\pi (\lambda | X_1, X_2, \cdots , X_ n)[/mathjaxinline].
Use SumXi for [mathjaxinline]\sum _{i=1}^ n X_ i[/mathjaxinline].
[mathjaxinline]\pi (\lambda | X_1, X_2, \cdots , X_ n) \propto[/mathjaxinline]
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