Binomial, normal and Poisson models
| English | 中文 | Pinyin |
|---|---|---|
| expected value/ekˈspektɪd ˈvæljuː/ | 期望值 | qī wàng zhí |
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.
Choose the mathematical structure
- 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. A Poisson model describes counts with a constant rate and appropriate independence assumptions.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines expected value?
The probability-weighted mean of a random variable.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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. A Poisson mean of 3 per hour gives mean 6 over two hours.
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).
No successes means five failures: 0.8⁵=0.32768.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every count of successes has a binomial distribution regardless of dependence. This claim is false. Explain which definition or assumption it violates.
Find Var(X) for X binomial(5,0.2).
Binomial variance=np(1-p)=5×0.2×0.8=0.8.
Every count of successes has a binomial distribution regardless of dependence.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For μ=100 and σ=15, find the z-score of 130.
Standardise: z=(130-100)/15=2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The probability-weighted mean of a random variable. Choose the relationship, show the method, check its assumptions and interpret the result.