Ultimately, observable events carry positive probabilities, making it impossible to directly witness negative probabilities. Negative probability is essentially a signed measure for $P$, rather than just a measure. However, this approach causes several commonly understood probability concepts to become ambiguous.

Solution:

The presence of probability associated with negative values depends on the distribution. For instance, when utilizing the standard normal distribution (which has a mean of 0 and variance of 1) to draw numbers, the probability of obtaining a negative value is 1/2.

The reason behind defining “general” formulas with integration from $-infty$ to $infty” is that density functions are always zero beyond their support. Hence, to calculate the expectation of the unit uniform, you need to integrate it.

The equation states that the integral of the product of x and f(x) from negative infinity to infinity is equal to the integral of the same product from 0 to 1, which is also equal to the integral of x from 0 to 1.

As $f$ is only defined on the interval $[0,1]$, it is equal to zero everywhere else.

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