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Symmetry Properties

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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1.6 Trigonometric Functions

Graphs of Sine and Cosine Functions

  • The graph of where -\pi<x<3\pi are plotted below. Do note that it extends beyond the drawn range.
  • There are a few observations we can make, and we can tie it back to what we have already learnt:

i) The graph repeats itself after an interval of 2\pi units. We say that the graph has a period of 2\pi, and hence is called a periodic function. Previously, we have learnt that \sin(x+2\pi).

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