The Normal Distribution [Video Tutorial]

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) $P(-Z<a)$

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 $P(X>8)=P(Z>b)$

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) $P(Y<b\,|\,Y<a)$

(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) $P(Y>1)$

(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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