为何某概率题中P(a<x<b)等于F(a)+F(b)而非常规F(b)-F(a)？

Great question—this almost always boils down to two key details about your problem's probability distribution and how the function F(x) is defined:

1. Your distribution has an even, symmetric PDF (about the origin) #

First, let's recall that for a random variable with a probability density function f(x) that's an even function (meaning f(-x) = f(x)), we have a critical symmetry property:

• P(X ≤ -k) = P(X ≥ k) for any positive k

This applies to common distributions like the standard normal, Cauchy, or uniform distribution over [-c, c].

2. The function F(x) is a survival function, not the standard CDF #

Here's the gotcha: many textbooks or problem sets will use F(x) to denote the survival function (P(X ≥ x)) instead of the standard cumulative distribution function (CDF, P(X ≤ x)).

Let's tie this together with an example that matches your scenario:
Suppose we want to calculate the two-tailed probability P(X < -a OR X > b) where a, b > 0.

• Using the survival function definition:
  • P(X > b) = F(b) (by survival function definition)
  • P(X < -a) = P(X ≥ a) = F(a) (thanks to the symmetry of the even PDF)
• Adding these gives exactly F(a) + F(b)—which is the calculation you're seeing in your problem.

If the problem explicitly states it's calculating P(a < x < b), there's likely a notation mix-up: maybe the interval is actually the complement of (a, b) (i.e., (-∞, a) ∪ (b, ∞)), or a and b are negative values that map to the positive tail via symmetry.

Quick sanity check with standard normal #

Let's use the standard normal distribution to verify:

• Standard CDF is Φ(x) = P(X ≤ x), survival function is 1 - Φ(x) = P(X ≥ x)
• If we calculate P(X < -1 OR X > 2):
  • P(X < -1) = Φ(-1) = 1 - Φ(1)
  • P(X > 2) = 1 - Φ(2)
• If we let F(x) = 1 - Φ(x) (survival function), this becomes F(1) + F(2)—perfectly matching the F(a)+F(b) pattern.

Bottom line #

The F(a)+F(b) calculation isn't replacing the standard F(b)-F(a) for interval probabilities—it's a specialized case for two-tailed tail probabilities when:

• The distribution is symmetric (even PDF about the origin)
• F(x) refers to the survival function (P(X ≥ x)) instead of the standard CDF.

内容的提问来源于stack exchange，提问作者WorldGov