The random variables X and Y are continuous. Is this enough information to determine the value of P(X² = e³)?

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The random variables X and Y are jointly continuous. Is this enough information to determine the value of P(X² = 3)?

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Answer 1

In the context of continuous random variables, the probability of a continuous random variable taking on a specific value is always zero. This is because the set of possible values for a continuous random variable forms a continuum, and the probability is defined over intervals.

1. For the first statement concerning \( P(X^2 = e^3) \): Since \( X \) is a continuous random variable, \( P(X^2 = e^3) = 0 \).

2. For the second statement regarding \( P(X^2 = 3) \): Similarly, since \( X \) is a continuous random variable, regardless of the relationship of \( X \) with \( Y \), \( P(X^2 = 3) = 0 \).

Thus, the answer to both statements is that it is not possible to determine a non-zero probability, and the value of both probabilities is 0.