Binomial, normal and Poisson models · Modelos binomial, normal e de Poisson
| English | Português |
|---|---|
| expected value/ekˈspektɪd ˈvæljuː/ | valor esperado |
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
- 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.
- 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.
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 · Modelos binomial, normal e de Poisson
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.
E(X)=0×0.2+1×0.5+2×0.3=1.1.
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.
A random variable with three possible values must be binomial. This claim is false. Explain which definition or assumption it violates.
Find its variance.
E(X²)=1.7, so variance=1.7-1.1²=0.49.
A random variable with three possible values must be binomial.
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
- S1 includes discrete random variables and the normal distribution. Named binomial and Poisson models are S2 content.
- 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
- edexcel IAL mathematics; official unit S1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- 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.