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# ACMMM169

## 3.5 Finding Normal Distribution Probabilities

#### Standardised Values and Finding Normal Distribution Probabilities

• Previously, we see that any normal distribution can be converted into a standard one via

\operatorname{Pr}(X \leq a)=\operatorname{Pr}\left(Z \leq \frac{a-\mu}{\sigma}\right) \equiv \operatorname{Pr}(Z \leq z)

where X \sim N(\mu, \sigma), and Z \sim N(0,1).

• Such expression is useful as we convert any values into one that is in terms of standard deviation(s) away from the mean. These new values are called standardised values or z-values. In particular we have

z=\frac{x-\mu}{\sigma} \text { or standardised value }=\frac{\text { data value }-\text { mean of normal curve }}{\mathrm{s} . \mathrm{d} . \text { of the curve }}

• Therefore, we can infer that: A positive z-value indicates that the data value it represents lies above the mean. If it is negative, then it is below the mean.
• Knowing how to convert any normal distribution to a standard one is important as it helps us to find the probabilities of any normal distributed events.
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## 3.4 The Normal Distribution

#### The Standard Normal Distribution

• The standard normal distribution has the following probability density function

f(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^{2}}

and the domain is R.

• The graph of f(x) is shown at the right. Notice that it is symmetric over x=0
• If a random variable X follows the standard normal distribution, then we would denote X \sim N(0,1).
• Note that standard normal distribution is a special case of normal distribution, with a mean 0 and variance 1. Therefore, this explains the N(0,1) written above.
• The standard normal distribution also have some important graphical features:
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