[Challenge] a novel normal distribution question

[Mao]

While doing some work, I came across a very interesting problem. It can be solved with methods knowledge, but it requires a very advanced level of understanding of probability and calculus. By that, I mean it relies on a good understanding of the fundamentals, but this would be unlike any other questions you've seen before. I have simplified the problem to a level manageable by the TI-89, and turned it into a worded question. I think there's a few of you who would appreciate this.

Give numerical answers. You will need to use your calculator.

Imagine in a simplified VCE system, you only have to do two subjects. English, and Methods. In this system, there is no VTAC scaling, and the ATAR is calculated by adding the study scores together. E.g. if you get 35 in English, and 30 in Methods, your ATAR is 65. In this simplified system, entry into Med requires an ATAR of 85, and an English score of 35.

1) Assume the study scores can be represented by normal distributions. SS for English has mean = 30, sd = 7. SS for methods has mean = 34, sd = 7. Typical normal distributions are unbounded, so there can be a very unlucky person out there with an SS of -10, and a very lucky person with an SS of 9001. So individual SS can go up to infinity, and go down to negative infinity. Let's treat it that way.

Under these assumptions, what is the probability that a randomly selected student will get into Med?

Hint: you will need to remember how pdfs work, and how to treat probabilities of independent events

2) It is not difficult to use 'truncated' normal distributions, that is normal distributions with boundaries imposed. So for SS, it would make sense that scores beyond 50 and below 0 are impossible. It is intuitive that such a truncation should be performed by using conditional probability. For example, for the English SS,



You will find wikipedia explains this at greater length, but is essentially saying the same thing (search for truncated normal distribution).

Using truncated normal distributions to describe the SS (i.e. truncation at 0 and 50 for both English and MM), what is the probability that a randomly selected student will get into Med?

Hint: expresse the truncated normpdf and normcdf in terms of normpdf, normcdf and constants. Take care when dealing with integration and cdf boundaries.


Good luck!


Also, special honors will go to anyone who can arrive at the answer by using random number generators. Will require some programming skills. (you will need ~100000 random numbers to get 2 sig fig precision, ~10 million random numbers to get 4 sig fig precision.)