This tutorial covers material encountered in chapter 16 of the VCE Mathematical Methods Textbook, namely:
- Normal random variables
- The standard normal distribution
- Standardising a normal distribution
Q1 – Calculate Probability for a Standard Normal Distribution
Q2 – Standardising a Normal Distribution
Q3 – Calculate Probability for a Normal Distribution with Mean and Standard Deviation Provided
Q4 – Find Probability for a Normal Distribution in Terms of the Standard Normal Distribution
Q5 – Application of the Normal Distribution
Worksheet
Q1. If \(Z\) is the standard normal distribution and \(P(Z\leq a)=p\), find in terms of \(p\):
(a) \(P(Z>a)\)
(b) \(P(-Z<a)\)
(c) \(P(-a<Z<a)\)
Q2. Let \(X\) be a normal distribution with mean 5 and standard deviation 3. Let \(Z\) be the standard normal random distribution. Find:
(a) \(a\) if \(P(X<2)=P(Z<a)\)
(b) \(b\) if \(P(X>8)=P(Z>b)\)
(c) \(P(X>5)\)
Q3. Let \(Y\) be a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). If \(\mu<a<b\) with \(P(Y<b)=p\) and \(P(Y<a)=q\), find:
(a) \(P(Y<a\,|\,Y<b)\)
(b) \(P(Y<b\,|\,Y<a)\)
(c) \(P(Y>b\,|\,Y>a)\)
(d) \(P(Y<2\mu – b)\)
Q4. Let \(Y\) be a normal distribution with mean 6 and standard deviation 3. Write each of the following in terms of the standard normal distribution \(Z\):
(a) \(P(Y<4)\)
(b) \(P(Y>1)\)
(c) \(P(1<Y<5)\)
(d) \(P(1<Y<10)\)
Q5. Suppose that Bill has recently completed four exams for his subjects, the results of which are shown below:
\begin{array}{c||ccc}
& \mu & \sigma & \text { Bill's mark } \\
\hline \text { Music Composition } & 65 & 14 & 85 \\
\hline \text { Japanese 1 } & 68 & 15 & 82 \\
\hline \text { Complex Analysis } & 55 & 16 & 68 \\
\hline \text { Metric and Hilbert Spaces } & 53 & 13 & 73
\end{array}Rank Bill’s performance in his exams relative to the other students from his best to his worst exam.
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