MATH / A2 level

Statistics option: inference and continuous models

Poisson distributions model counts under constant-rate, independent-event assumptions. A probability density integrates to probability; its height alone is not a point probability.

Learning objectives

Explain statistics option: inference and continuous models using a concrete example.
Apply the worked method and justify each step.
Recognise the limitation described in the final check.

The big idea

Poisson distributions model counts under constant-rate, independent-event assumptions. A probability density integrates to probability; its height alone is not a point probability.

Sampling distributions quantify random variation in estimates. Confidence intervals and significance tests answer different questions; specify hypotheses, tail and significance level before interpreting a p-value.

Equations and relationships

P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!}

Poisson probability for r=0,1,2,… .

Worked example

For X~Poisson(2), find P(X=0).

Follow the solution

Substitute r=0: e^−2×2⁰/0!=e^−2.

Investigate and explain

Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.

Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.

Open the interactive investigation

Check your understanding

1. For a continuous random variable, P(X=a) is what?

0
The density at a
1

Check the answer and reasoning

A single point has zero width and hence zero area under the density.

2. For X~Poisson(2), find P(X=0).

0
e^−2
2e^−2

Check the answer and reasoning

e^−2

Substitute r=0: e^−2×2⁰/0!=e^−2.

3. Does a small p-value prove the alternative hypothesis?

Yes
Only at 5%
No

Check the answer and reasoning

It measures incompatibility with the null under assumptions, not the probability that a hypothesis is true.