Binomial, normal and Poisson models
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| expected value/ekˈspektɪd ˈvæljuː/ | giá trị kỳ vọng |
What makes a count predictable?
- A quality inspector counts defective items. The number is random, but a model can describe its likely range.
- This lesson studies expected value 期望值: The probability-weighted mean of a random variable.
The method
- A binomial model needs fixed n, independent trials, two outcomes and constant p.
- E(X)=np and Var(X)=np(1-p).
- For a normal model use z=(x-μ)/σ.
- State the event before calculating a tail probability.
A count of successes fails the binomial model when which condition breaks?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The binomial model needs fixed n, independent trials, two outcomes and a constant p. Sampling without replacement makes trials dependent, so the binomial formula no longer applies exactly.
A worked case
For X binomial(5,0.2), P(X=0)=0.8^5=0.32768, E(X)=1 and Var(X)=0.8. For a normal quantity with μ=100,σ=15, the value 130 has z=2. Poisson is not part of this AQA 7357 lesson.
Binomial, normal and Poisson models
A binomial model needs fixed n, independent trials, two outcomes and constant p
Compare the model with the worked case and explain one change.
For X binomial(5,0.2), find P(X=0). Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
No successes means five failures: 0.8⁵=0.32768.
Where it goes wrong
- Not every count is binomial: changing p or dependence can invalidate it.
- For a continuous variable, the probability of one exact value is zero.
- Continuity correction matters when approximating a discrete distribution by a normal one.
Not true: Every count of successes has a binomial distribution regardless of dependence. The binomial model needs independent trials with a fixed probability; dependent trials break it.
Find Var(X) for X binomial(5,0.2).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Binomial variance=np(1-p)=5×0.2×0.8=0.8.
Every count of successes has a binomial distribution regardless of dependence.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The binomial model needs independent trials with a fixed probability; dependent trials break it.
Applying it
- Write the event as an inequality before using calculator distribution functions.
- Distinguish P(X<k), P(X≤k) and a tail complement.
- State assumptions in context.
For μ=100 and σ=15, find the z-score of 130.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Standardise: z=(130-100)/15=2.
Match each distribution fact to its formula or statement.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The binomial moments follow from n and p; the z-score converts any normal value into standard units for table use.
In your course
- AQA A-level Mathematics (7357), specification section N.
- Expected value: The probability-weighted mean of a random variable.
- A binomial model needs fixed n, independent trials, two outcomes and constant p.
- Watch for: not every count is binomial: changing p or dependence can invalidate it.