Normal Distribution Summary

Distribution Function

  • F(x) = P(X ≤ x)

    • Describes the probability that a random variable X is less than or equal to x.

Normal Distribution

  • A family of distributions characterized by mean \mu and standard deviation \sigma.

  • Standard Normal Distribution:

    • Mean of 0 and standard deviation of 1.

    • If X is normally distributed with mean \mu and standard deviation \sigma, then Z = (X - \mu) / \sigma has a standard normal distribution.

    • Conversely, if Z is a standard normal, X = \mu + \sigma Z is a normal distribution with mean \mu and standard deviation \sigma.

  • Notation: If X is normally distributed with mean \mu and standard deviation \sigma, we write X \sim N(\mu, \sigma^2).

    • (X + a) \sim N(\mu + a, \sigma^2)

    • (X/b) \sim N(\mu/b, \sigma^2/b)

    • (X - \mu)/\sigma \sim N(0, 1)

Central Limit Theorem

  • If X1, X2, …, X_n are independent samples from a distribution, the distribution of the sample mean \bar{X} approaches normal as n tends to infinity.

Sampling Distribution

  • If data are normally distributed with population mean\mu and standard deviation \sigma, then the sample mean \bar{x} \sim N(\mu, \sigma^2/n).

  • Even if the data are NOT normally distributed, provided the sample is large, the above still holds approximately.

Normal Probabilities

  • If Z \sim N(0, 1), then probabilities can be calculated using the distribution function.

  • Example: If P(Z < z0) = 0.950, find z0 using tables or a computer program.

  • P(Z > -2) = 1 - P(Z < -2)

  • P(-2 < Z < 3) = P(Z < 3) - P(Z < -2)

Distribution Function

  • F(x) = P(X ≤ x)- Describes the probability that a random variable X is less than or equal to x.

Normal Distribution

  • Family of distributions: mean \mu, standard deviation \sigma.

  • Standard Normal Distribution: Mean 0, standard deviation 1.

    • If X is normally distributed with mean \mu and standard deviation \sigma, then Z = (X - \mu) / \sigma has a standard normal distribution.

    • Conversely, if Z is a standard normal, X = \mu + \sigma Z is a normal distribution with mean \mu and standard deviation \sigma.

  • Notation: X \sim N(\mu, \sigma^2).

    • (X + a) \sim N(\mu + a, \sigma^2)

    • (X/b) \sim N(\mu/b, \sigma^2/b)

    • (X - \mu)/\sigma \sim N(0, 1)

Central Limit Theorem

  • Sample mean \bar{X} approaches normal as n tends to infinity.

Sampling Distribution

  • If data distributed with population mean\mu and standard deviation \sigma, then the sample mean \bar{x} \sim N(\mu, \sigma^2/n).

  • Even if the data are NOT normally distributed, provided the sample is large, the above still holds approximately.

Normal Probabilities

  • If Z \sim N(0, 1), probabilities calculated using distribution function.

  • Example: If P(Z < z0) = 0.950, find z0 using tables or a computer program.

  • P(Z > -2) = 1 - P(Z < -2)

  • P(-2 < Z < 3) = P(Z < 3) - P(Z < -2)