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
- For a discrete random variable, sum the probabilities to 1 and calculate E(X)=sum xP(X=x).
- For the normal model use z=(x-μ)/σ and calculate an appropriate area.
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
If X takes 0,1,2 with probabilities 0.2,0.5,0.3, E(X)=0×0.2+1×0.5+2×0.3=1.1. Also E(X²)=0+0.5+1.2=1.7, so Var(X)=1.7-1.1²=0.49. For μ=100,σ=15, a value of 130 has z=2.
Binomial, normal and Poisson models
For a discrete random variable, sum the probabilities to 1 and calculate E(X)=sum xP(X=x)
Compare the model with the worked case and explain one change.
Find E(X) for values 0,1,2 with probabilities 0.2,0.5,0.3.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
E(X)=0×0.2+1×0.5+2×0.3=1.1.
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: A random variable with three possible values must be binomial. The binomial model needs independent trials with a fixed probability; dependent trials break it.
Find its variance.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
E(X²)=1.7, so variance=1.7-1.1²=0.49.
A random variable with three possible values must be binomial.
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
- S1 includes discrete random variables and the normal distribution.
- Named binomial and Poisson models are S2 content.
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
- Pearson Edexcel International A Level Mathematics, unit S1.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Expected value: The probability-weighted mean of a random variable.
- For a discrete random variable, sum the probabilities to 1 and calculate E(X)=sum xP(X=x).
- Watch for: not every count is binomial: changing p or dependence can invalidate it.
Write from a density constant of 1/8
- f of x equals k x on the interval from 0 to 4, and the total area is 1.
- Find k.
f of x equals k x for x from 0 to 4, and the area under the graph is 1. What is k?
Half of 16 is 8, and 8 k equals 1.
What is k?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The area condition forces k to be 1/8. Other options — 1/2 (the lesson value of k); 4 (the upper end of the interval).