假设随机变量的 PDF 如下：

P(X

Question

假设随机变量的 PDF 如下：

P(X < 0)的值是多少？

Suppose that the PDF of a random variable is as follows:

What is the value of P(X < 0)?

BlackTom AI · Accepted Answer

The probability P(X < 0) is the integral of the PDF over (-∞, 0).

1) Relevant concepts
- For a continuous random variable with PDF f(x), P(a < X < b) = ∫_a^b f(x) dx.
- If f(x) = 0 on an interval, the integral over that interval is 0.
- A point mass at x = 0 (if it existed) would affect P(X ≤ 0) but not P(X < 0).

2) Step-by-step reasoning
- The problem statement (and the displayed PDF) indicates f(x) = 0 for every x < 0.
- Therefore P(X < 0) = ∫_{-∞}^{0} f(x) dx.
- Substitute f(x) = 0 on (-∞,0) into the integral.

3) Calculation (all intermediate steps)
- P(X < 0) = ∫_{-∞}^{0} f(x) dx
- = ∫_{-∞}^{0} 0 dx
- = 0

4) Why the correct answer is correct
- Since the PDF takes the value zero at all points left of 0, there is no probability mass assigned to any subinterval contained in (-∞,0). Integrating zero over any interval yields zero probability.

5) Why the other choices are incorrect
- 1 would mean all probability is concentrated strictly to the left of 0, which contradicts f(x)=0 for x<0.
- 0.5 or 2/3 would assert a nonzero fraction of total probability lies left of 0; again impossible because the PDF is identically zero there.
- Any nonzero probability on (-∞,0) would require f(x)>0 on some subset of (-∞,0), which is not the case.

6) Conclusion
- Therefore P(X < 0) = 0, which matches the provided answer 0.

假设随机变量的 PDF 如下： P(X < 0)的值是多少？ Suppose that the PDF of a random variable is as follows: What is the value of P(X < 0)?单项选择题

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