Proof of bay s theorem

In mathematics, we will study different theorem. Here we will understand the concept of bay s theorem. Bay’s theorem can have two distinct interpretations. It is an important concept of Bayesian statistics and has different properties in the field of science and engineering. Formula that is used to solve the probability, that name was given after the 18 th – century by the great scientist British mathematician Thomas bayes. Now we talk about the formula used in Bay’s theorem which is given below :
K (I / J) = K (I ∩ J) / K (J) = K (I) * K (J | I) / K (J)
Now we will understand the proof of bay’s theorem: Suppose we have X and Y_j be two sets. Then the conditional probability requires that:
P (X ∩ Y_j) = P (X) P (Y_j | P), here the symbol ‘∩’ represented as intersection (‘and’) and also said :
P (X ∩ Y_j) = P (Y_j ∩ X) = P (Y_j) P (X | Y_j), therefore it can be written as:
P (Y_j ∩ X) = P (Y_j) P (X | Y_j) / P (X), now assume Z = ∪_{i = 1}ⁿX_i, so X_i is an event in Z and X_i ∩ X_j = ⱷ for i ≠ j, then we can write it as:
X = X ∩ Z = X ∩ (∪_{i = 1}ⁿX_i) = ∪_{i = 1}ⁿ(X ∩ X_i),
P (X) = P (∪_{i = 1}ⁿ(X ∩ X_i)) = ∑_{i = 1}^N P (X ∩ X_i), it can also be written as:
P (X) = ∑_{i = 1}^N P (X_i) P (X | X_i),
P (X_i | X) = P (X_i) P (X | X_i) / ∑_{i = 1}^N P (X_i) P (X | X_i) this is the proof of bay’s theorem.
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