# The Normal Distribution

## The Normal Distribution Revision

**The Normal Distribution**

The **normal distribution** is a **bell shaped curve** that is **symmetric about the mean**. It is defined by its **mean** \mu and its **standard deviation** \sigma. If X is **normally distributed**, we write X\sim N(\mu,\sigma^{2}).

**Facts About the Normal Distribution**

Probability is represented by the **area under the graph** (note this means the total area under the graph is 1).

The **normal distribution** is **symmetric about the mean** \mu, which means:

\mathbb{P}(X\geq\mu)=\mathbb{P}(X\leq\mu)=0.5

\mathbb{P}(X\geq\mu+a)=\mathbb{P}(X\leq\mu-a)\text{ for all }a

The **standard deviation** determines how flat the graph is – the** higher the standard deviation, the lower the graph**.

The graph** tends to 0** as it travels away from the mean, but it **never gets there**.

**Using a Calculator for the Normal Distribution**

Your **calculator** might have a built in function for **normal distribution** probabilities.

This will ask you to put in a **mean** and **standard deviation** and an **upper and lower bound** on X to produce a probability.

This is fairly straightforward for questions such as \mathbb{P}(3\leq X\leq 6), but what about questions such as \mathbb{P}(X\leq 5), where there is no **lower bound**?

In this case, take the** lower bound** to be as low as your calculator allows you to input, so that it has **as little effect on the result as possible**.

Usually, -9999 for a **lower bound** or 9999 for an **upper bound** will suffice.

**The Inverse Normal Function**

Sometimes, you might be given a probability p and asked to find x such that p=\mathbb{P}(X<x).

For this, we use the **inverse normal function**, which should **also be on your calculator**.

You will input a **mean**, **standard deviation** and probability and the calculator gives you x such that p=\mathbb{P}(X<x).

**Note:** For < and \leq you can do this directly on your calculator. However, for > and \geq you need to subtract p from 1 to turn P(X > a) into P(X \leq a) = 1-p, then you can use your calculator as normal.

**Example: The Normal Distribution**

In the 2012 Olympics Men’s 100 metre sprint final, the average time taken was 10 seconds. The times were **normally distributed** with a variance of 0.2 seconds. Calculate the probability of a runner having finished the race in 9.58 seconds or less.

**[2 marks]**

Use your calculator, upper bound 9.58 and lower bound -9999

\mathbb{P}(X\leq 9.58)=0.1738The probability of a runner finishing the race in 9.58 seconds or less is 0.1738

## The Normal Distribution Example Questions

**Question 1: **For X\sim N(4,1), calculate:

a) \mathbb{P}(X\leq 3.5)

b) \mathbb{P}(X\geq 6)

c) \mathbb{P}(X\leq 4.25)

**[3 marks]**

**Question 2: **For X\sim N(75,100), calculate:

a) \mathbb{P}(90\leq X\leq 110)

b) \mathbb{P}(65\leq X\leq 75)

c) \mathbb{P}(50\leq X\leq 100)

**[3 marks]**

**Question 3: **For X\sim N(20,25), find the value of a such that:

a) \mathbb{P}(X>a)=0.1

b) \mathbb{P}(X\leq a)=0.6

c) \mathbb{P}(15\leq X\leq a)=0.4

**[4 marks]**

**Question 4: **The weights of punnets of strawberries sold by a greengrocers are 250\text{ g} on average with a standard deviation of 9\text{ g}. What is the probability that Jenny gets a 280\text{ g} punnet or better from the grocers?

**[2 marks]**

We model this problem with X\sim N(250,81).

We want to find \mathbb{P}(X\geq 280)

\begin{aligned}\mathbb{P}(X\geq 280)&=1-\mathbb{P}(X\leq 280)\\[1.2em]&=1-\mathbb{P}(X\leq 280)\\[1.2em]&=1-0.9996\\[1.2em]&=0.0004\end{aligned}