## 5.6Optional: Working with normal distributions

This question is optional; for example, if you need more practice, or you are studying for the exam.

(Answers are available in Sect. A.6)

This question has a video solution in the online book, so you can hear and see the solution.

The Southern Oscillation Index (SOI) is a unitless2 climatological measure that is easily computed, and has been shown to be related to the weather conditions in eastern Australia .

The daily SOI has an approximate normal distribution, and is designed to have a mean of 0 and a standard deviation of 10.

1. Draw a rough, labelled sketch of the theoretical distribution of the SOI values.
2. What is the probability that the daily SOI exceeds $$20$$?
3. What are the odds that the daily SOI exceeds 20?
4. What is the probability that the daily SOI is less than $$-25$$?
5. What is the probability that the daily SOI is greater than $$-12$$?
6. What is the probability that the daily SOI is between $$-10$$ and $$20$$?
7. The SOI is less than what value about 80% of the time?
8. The SOI is greater than what value about 35% of the time?

### References

Dunn PK. Bootstrap confidence intervals for predicted rainfall quantiles. International Journal of Climatology. 2001;21(1):89–94.
Stone RC, Auliciems A. soi phase relationships with rainfall in eastern Australia. International Journal of Climatology. 1992;12:625–36.