求解随机变量x∈(6,14)的概率：已知P(x>6)与P(x<16)如何应用？

Hey there, let's walk through why your initial approach didn't work and what's needed to solve this properly.

First, let's clarify what we can derive from the given values:

• P(x > 6) = 0.9095 → This means P(x ≤ 6) = 1 - 0.9095 = 0.0905 (since total probability across all outcomes equals 1)
• P(x < 16) = 0.8360 → This means P(x ≥ 16) = 1 - 0.8360 = 0.1640

Your attempt to divide these probabilities doesn't apply here—probability division is only used for conditional probability scenarios (like calculating P(A|B) = P(A∩B)/P(B)), which isn't what we're trying to find with this range probability.

The Gap in Existing Information #

To compute P(6 < x < 14), we need to calculate P(x < 14) - P(x ≤ 6)—this subtracts the probability of x being ≤6 from the probability of x being <14 to isolate the middle range we care about.

The problem is: we don't have P(x < 14), and without knowing the underlying distribution of x (e.g., normal, uniform, exponential), we can't infer this value from the two given probabilities alone.

Example: If x Follows a Normal Distribution #

If we assume x is normally distributed (a common real-world scenario), we can use the known probabilities to estimate the mean (μ) and standard deviation (σ), then calculate the desired range:

1. Convert the known probabilities to Z-scores using the inverse normal distribution:
   • P(x ≤ 6) = 0.0905 corresponds to a Z-score of ~-1.33 (the 9.05th percentile of the standard normal distribution)
   • P(x < 16) = 0.8360 corresponds to a Z-score of ~0.98 (the 83.6th percentile of the standard normal distribution)
2. Set up equations for Z-scores:

   (6 - μ)/σ = -1.33
   (16 - μ)/σ = 0.98
   

3. Solve for μ and σ:
   Subtract the first equation from the second: (16-6)/σ = 0.98 - (-1.33) → 10/σ = 2.31 → σ ≈ 4.33
   Plug σ back into the first equation: 6 - μ = -1.33*4.33 → μ ≈ 6 + 5.76 = 11.76
4. Calculate the Z-score for x=14: (14 - 11.76)/4.33 ≈ 0.52
5. Find P(x < 14) by looking up the Z-score 0.52 in the standard normal table (~0.6985)
6. Finally, P(6 < x < 14) = 0.6985 - 0.0905 = 0.608

Key Takeaway #

Without knowing the distribution of x, or having an additional probability value (like P(x < 14)), you can't compute the exact probability you're looking for. The critical missing piece is information about how x is distributed.

内容的提问来源于stack exchange，提问作者Lara Jane