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​​Discussion 12A

​​Normal Distribution

​​​​$X \sim \mathcal{N}(\mu, \sigma^2)$​

​​​​$f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{\frac{-(x-\mu)^2}{2 \sigma^2}}$​

​​​​$X$​ has mean ​​$\mu$​ and variance ​​$\sigma^2$​.

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​​​​$Z \sim \mathcal{N}(0,1)$​ has the standard normal distribution.

​​If ​​$X = a Z + b$​ then ​​$X \sim \mathcal{N} (b, a^2)$​.

​​If ​​$X\sim \mathcal{N}(\mu_X, \sigma_X^2)$​ and ​​$Y \sim \mathcal{N} (\mu_Y, \sigma_Y^2)$​ are independent then ​​$X + Y \sim \mathcal{N} (\mu_X + \mu_Y, {\sigma_X^2 + \sigma_Y^2})$​.

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​​The z-score of a sample point is its position from the mean, in number of standard deviations.

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​​The CDF of the normal distribution is ​​$\Phi(z) = \Pr(Z < z)$​. There is no closed form for the CDF, so we use a table of approximations. The CDF maps from z-scores to probabilities, while the inverse CDF maps from probabilities to z-scores.

​​Hypothesis Testing

1. ​​Formulate a null hypothesis ​​$H_0$​ which says that a setting is ordinary

2. ​​Formulate an alternative hypothesis ​​$H_1$​ which says that a setting is unusual

3. ​​Run an experiment and record an observation

4. ​​Calculate the p-value of the outcome: the probability that the outcome (or a more extraordinary outcome) occurs assuming the null hypothesis ​​$H_0$​ is true

5. ​​If the p-value less than a certain value (often ​​$0.05$​) then the observation is considered statistically significant and we reject ​​$H_0$​.

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