Binomial Distribution B(n,p)
E(X)=np、V(X)=np(1−p)
Example ①
The number of heads X when a coin is tossed 10 times, X∼B(10,1/2): E(X)=5、σ(X)=√2.5≒1.58
The number of heads X when a coin is tossed 10 times, X∼B(10,1/2): E(X)=5、σ(X)=√2.5≒1.58
Normal Distribution N(μ,σ²)
Standardization: Z=(X−μ)/σ ∼ N(0,1)
Approximately 68% lies within μ±σ, and approximately 95% lies within μ±2σ
Example ②
For N(60,10²), the proportion scoring at least 70 points (P(0≤Z≤1)≒0.341):
Z=(70−60)/10=1 → P(X≥70)=0.5−0.341=0.159 (approximately 16%)
For N(60,10²), the proportion scoring at least 70 points (P(0≤Z≤1)≒0.341):
Z=(70−60)/10=1 → P(X≥70)=0.5−0.341=0.159 (approximately 16%)
Key Points
- Binomial distribution → E=np、V=np(1-p)
- When n is large, B(n,p)≒N(np,np(1-p))
Practice Problems
- Find E(X) and V(X) for X∼B(20,0.3).
- For N(50,5²), what is the probability of being between 40 and 60? (P(0≤Z≤2)≒0.477)
- For X∼B(100,0.5), what is the probability of X≥60? (Use the normal approximation, P(0≤Z≤2)≒0.477)
Answers
- E(X)=6、V(X)=4.2
- P(−2≤Z≤2)≒0.954
- μ=50、σ=5。Z=(60-50)/5=2 → P(X≥60)≒0.023