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 $X<em>1, X</em>2, …, X_n$ 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 $P(Z < z<em>0) = 0.950$, find $z</em>0$ 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 $P(Z < z<em>0) = 0.950$, find $z</em>0$ using tables or a computer program.

• $P(Z > -2) = 1 - P(Z < -2)$

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