uni/year2/semester1/logseq-stuff/pages/The Normal Distribution.md

• #ST2001 - Statistics in Data Science I
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• What is a Normal Distribution? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:07:27.457Z card-last-score:: 1 id:: 63510f7d-d646-41b8-82d7-8634c840892e
  • A random variable X with probability distribution function
    • f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{(x - \mu)^2}{2 \sigma^2}} -\infty < x < \infty
    • is a normal random variable with parameters \mu & \sigma (where \infty < \mu < \infty and \sigma > 0) where \mu is the mean and \sigma is the standard deviation.
  • Write X \sim N(\mu, \sigma^2).
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• Features of the Normal Distribution #card

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  • Also called the Gaussian Distribution.
  • The pdf (probability density function) is a bell-shaped curve.
  • The distribution of many types of observations can be approximated by a Normal Distribution.
  • Single mode.
  • Symettric.
  • Model for continuous measurements.

• Examples of Normal Distributions

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  • Normal Curves with \mu_i = \mu_2 and \sigma_1 < \sigma_2

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  • Normal Curves with \mu_1 < \mu_2 and \sigma_1 < \sigma_2

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• Empirical Rule for a Normal Distribution #card

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  • For any normal random variable:
    • P(\mu - \sigma < X < \mu + \sigma) = 0.6827
    • P(\mu - 2\sigma < X < \mu + 2\sigma) = 0.9545
    • P(\mu - 3\sigma < X < \mu + 3\sigma) = 0.9973
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  • The 68-95-99.7 Rule

    • Normal models give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean.
    • In a normal model:
      • About 68% of the values within one standard deviation from the mean.
      • About 95% of the values fall within two standard deviations of the mean.
      • About 99.7% of the values fall within three standard deviations of the mean.

• Areas Under a Normal Curve #card

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  • To calculate a probability in a range under a normal distribution.
    • P(x_1 < X < x_2) = \int_{x_1}^{x_2} \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{(x - \mu)^2)}{2\sigma^2}}dx
  • For example. P(x_1 < X < x_2) = area of the shaded region.
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• z-scores

  • What is a z-score? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-16T00:00:00.000Z card-last-reviewed:: 2022-11-15T18:42:43.077Z card-last-score:: 1
    • A z-score reports the number of standard deviations from the mean.
    • For example, a z-score of 2 indicates that the observation is two standard deviations above the mean.

  • Converting to z-scores #card

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    • To convert a random variable X which follows a N(\mu, \sigma^2) to a random variable Z that follows a standard normal N(0,1), calculate Z as:
      • Z = \frac{X - \mu}{\sigma}
    • Convert X \sim N(100,100) to a random variable Z such that Z \sim N(0,1).

• Cumulative Distribution Functions

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  • How is the cumulative distribution function of a standard normal random variable denoted? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-19T00:00:00.000Z card-last-reviewed:: 2022-11-18T18:34:54.021Z card-last-score:: 1
    • The cumulative distribution function of a standard normal random variable is denoted as \Phi(z) = P(Z \leq z)

• Normal Approximation to the Poisson

  • If X is a Poisson random variable with E(X) + \lambda and V(X) = \lambda,
    • Z = \frac{X-\lambda}{\sqrt{\lambda}}
  • The approximation is good for \lambda \geq 5.

• Continuity Correction

  • Using the Normal Distribution to approximate a discrete distribution (e.g., Binomial) we need to take into account the fact that the Normal Distribution is continuous.
    • | \textbf{Discrete} | | \textbf{Continuous} | | P(X > k) | \rightarrow | P(X > k + \frac 1 2) | | P(X \geq k) | \rightarrow | P(X > k - \frac 1 2) | | P(X < k) | \rightarrow | P(X < k - \frac 1 2) | | P(X \leq k) | \rightarrow | P(X < k + \frac 1 2) | | P(k_1 < X < k_2 | \rightarrow | P(k_1 + \frac 1 2 < X < k_2 - \frac 1 2) | | P(k_1 \leq X \leq k_2) | \rightarrow | k_1 - \frac 1 2 < X < k_2 + \frac 1 2 |