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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Into the Normal Distribution

### Why do this problem?

This problem is based around understanding the probability density function for the normal distribution. The aim is to draw the learner into an understanding of the properties of pdfs without requiring too many complicated calculations: it uses and will reinforce ideas about functions, integration and areas and the use
of tables to calculate the probabilities for standardised normal distributions. It will also suit self-motivated independent learners.

### Possible approach

### Key questions

### Possible extension

### Possible support

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### Scale Invariance

### Into the Exponential Distribution

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This question could sensibly be used once students are starting to learn about the use of normal distribution tables and standardised normal distributions. There is a lot of scope for numerical estimation of probabilities and the first part could be used to reinforce the fact that a probability density function tells us quite a lot about a distribution even without the need for complicated
calculation. It will tie in nicely with other parts of the syllabus on numerical integration.

What do you know about the area under a pdf?

What does the area under a pdf between two points mean?

How might we write down our probabilistic statements in terms of standardised normal variables?

Can learners find any of the areas enclosed by the lines in the diagrams (using normal distribution tables).

Can they find the points of intersection on the graph?

Encourage learners to rely on their intuitive underestanding of integration in terms of area. Alternatively, focus on the last two parts of the question as a discussion. If they can't come up with their own suggestions of calculation, perhaps they might initially check the estimates of others?

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Get into the exponential distribution through an exploration of its pdf.