Expectation of X_t When Pr(X_t>b)=0: Finite?

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The discussion centers on the relationship between the probability Pr(X_t > b) = 0 and the expectation E(X_t) of a random variable X_t. It is established that if X_t is constrained such that 0 ≤ X_t ≤ b, then the expectation E(X_t) must also be finite and less than or equal to b. However, there are scenarios where the expectation can be infinite, indicating that the conditions of the random variable must be carefully considered.

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St41n
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If i know that Pr(X_t>b)=0, where X_t>0 and b is positive finite, then should the expectation of X_t be finite? Is there any case where it is infinite?
 
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Yes. Proof is trival. 0<=X _t <=b implies E(X_t)<=b.
 

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