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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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The expectation of X_t when Pr(X_t>b)=0: Finite refers to the expected value of a random variable X_t when the probability of X_t being greater than a given value b is equal to 0. This implies that the likelihood of X_t being greater than b is very low or practically impossible.
The expectation of X_t in this scenario is calculated by taking the integral of X_t from negative infinity to b, multiplied by the probability density function of X_t. This essentially means that the expectation is calculated by considering all possible values of X_t that are smaller than b, as the probability of X_t being greater than b is 0.
The probability of X_t being greater than b being equal to 0 means that the expected value of X_t is entirely determined by the values of X_t that are smaller than b. This implies that any values of X_t that are greater than b have no impact on the overall expectation, as their probability is negligible.
No, the expectation of X_t cannot be a negative value when Pr(X_t>b)=0: Finite. This is because the expected value is calculated by taking the integral of X_t from negative infinity to b, which implies that all values of X_t included in the calculation are positive or 0.
If the expectation of X_t is 0 when Pr(X_t>b)=0: Finite, it means that the expected value of X_t is equal to 0. This could occur if the probability density function of X_t is heavily skewed towards smaller values, or if the values of X_t that are greater than b have a very low probability of occurring.