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# The Normal Distribution

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

## 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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