A continuous random variable with a symmetric, bell-shaped distribution is said to follow the normal distribution N(μ, σ²).
Probability density function: f(x) = (1/(σ√(2π)))·exp(−(x−μ)²/(2σ²))
- Mean μ, variance σ², standard deviation σ
- Approximately 68% lies within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ
Example ① For N(50, 100) (σ=10), estimate the probability that 40 ≦ X ≦ 60.
Solution:This is the range μ±σ, so approximately 68%
Solution:This is the range μ±σ, so approximately 68%
Key Point
- The 68-95-99.7 rule (empirical rule)
- Many natural phenomena and measurement errors follow a normal distribution
Practice Problems
- For N(0, 1), estimate the probability that −2 ≦ X ≦ 2.
Solutions and Explanations
- Solution:approximately 95%