<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1778881128668-miil2hrscr9.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ul class="bullet-list"><li><div class="list-item-paragraph">Bell-shaped and symmetrical about \(x=\mu\).</div></li><li><div class="list-item-paragraph">Maximum value of the curve occurs when&nbsp;\(x=\mu\), therefore <strong>mode</strong> \(=\) <strong>median</strong> \(=\) <strong>mean</strong> \(=\mu\).</div></li><li><div class="list-item-paragraph">Points of inflexion (non-stationary) are at \(x=\mu-\sigma\) and at \(x=\mu+\sigma\).</div></li><li><div class="list-item-paragraph">Probability that the normal random variable \(X\) is within \(1\) standard deviation of the mean, i.e.<br> \[P(|X-\mu|&lt;\sigma)=0.683\] <br>or <br>\[P(\mu-\sigma&lt;X&lt;\mu+\sigma)=0.683\].</div></li><li><div class="list-item-paragraph">Also, we have&nbsp;\(P(|X-\mu|&lt;2\sigma)=0.954\) and \(P(|X-\mu|&lt;3\sigma)=0.998\).</div></li><li><div class="list-item-paragraph">The range of the distribution is therefore approximately \(6\) standard deviations.</div></li><li><div class="list-item-paragraph">The area under the curve from \(x=a\) to \(x=b\) represents the probability \(P(a \leq x \leq b)\).</div></li></ul>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1778881379203-6cschnl98jl.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Bell-shaped and symmetrical about \(x=\mu\).

Maximum value of the curve occurs when \(x=\mu\), therefore **mode** \(=\) **median** \(=\) **mean** \(=\mu\).

Points of inflexion (non-stationary) are at \(x=\mu-\sigma\) and at \(x=\mu+\sigma\).

Probability that the normal random variable \(X\) is within \(1\) standard deviation of the mean, i.e.
 \[P(|X-\mu|<\sigma)=0.683\] 
or 
\[P(\mu-\sigma<X<\mu+\sigma)=0.683\].

Also, we have \(P(|X-\mu|<2\sigma)=0.954\) and \(P(|X-\mu|<3\sigma)=0.998\).

The range of the distribution is therefore approximately \(6\) standard deviations.

The area under the curve from \(x=a\) to \(x=b\) represents the probability \(P(a \leq x \leq b)\).

Properties of the normal curve

Properties of the normal curve