Standard Normal Distribution – Notes

Setup of the normal model

• The variable X is modeled by a normal distribution with mean and standard deviation: $X \sim N(\mu,\sigma^2)$ with given values $\mu=100, \sigma=15$.
• The phrase "tilde" in the transcript indicates a distribution, i.e., X is distributed as a normal distribution: $X \sim N(100, 15^2)$.
• The goal is to work with the standard normal distribution, often denoted as Z, which is a standardized form of X.

Standard Normal distribution (Z distribution)

• Define the standardization: $Z = \frac{X - \mu}{\sigma}$.
• If X ~ N(\mu, σ^2), then $Z ~ N(0,1)$, the standard normal distribution.
• The standard normal has:
  • mean: $\mathbb{E}[Z] = 0$
  • standard deviation: $\operatorname{sd}(Z) = 1$
• In this context, the standard normal distribution is also referred to as the z-distribution.

Cumulative distribution for the standard normal

• The standard normal CDF is denoted as $\Phi(z) = P(Z \le z)$.
• Tables (or software) are used to look up values of $\Phi(z)$ for given z.
• To relate back to X, use the transformation:
  $P(X \le x) = P\Big(Z \le \frac{x-\mu}{\sigma}\Big) = \Phi\Big(\frac{x-\mu}{\sigma}\Big).$

Important properties and interpretations

• Symmetry and tail areas:
  • The total area under the standard normal curve is 1:
    $\int_{-\infty}^{\infty} \phi(z) \, dz = 1.$
  • Area to the left of the mean (z = 0) is 0.5:
    $P(Z \le 0) = \Phi(0) = 0.5.$
  • For symmetric distributions, the area to the right of z is the same as the area to the left of -z:
    P(Z > z) = P(Z < -z) = \Phi(-z).
• Complement rule and tail shading:
  • The area to the right of a positive z is the complement of the left-tail:
    P(Z > z) = 1 - \Phi(z).
  • This is also equal to the left-tail beyond the negative z by symmetry: P(Z > z) = \Phi(-z).

Concrete numerical examples (using the standard normal table)

• Example 1: z = 1.28
  • Left-tail: $\Phi(1.28) \approx 0.8997.$
  • Right-tail: P(Z > 1.28) = 1 - \Phi(1.28) \approx 0.1003.
  • Area above 1.28 equals the area below -1.28 by symmetry: P(Z > 1.28) = P(Z < -1.28) = \approx 0.1003.
• Example 2: z = 1.88
  • Left-tail: $\Phi(1.88) \approx 0.9699.$
  • Right-tail: P(Z > 1.88) = 1 - \Phi(1.88) \approx 0.0301.
  • Area above 1.88 equals area below -1.88 by symmetry: P(Z > 1.88) = P(Z < -1.88) \approx 0.0301.
• Example 3: z = 0
  • Left-tail: $\Phi(0) = 0.5.$
  • This reinforces that the mean (and median) of the standard normal is at z = 0.

Connecting to the original scale (X)

• Suppose we want P(X ≤ x). Compute the corresponding z-score and use the standard normal:
  $P(X \le x) = \Phi\left( \frac{x - \mu}{\sigma} \right).$
  For example, with x such that the z-score is 1.28, the probability is $\Phi(1.28) \approx 0.8997$.

Conceptual notes and practical tips

• Always visualize by drawing the normal curve and shading the appropriate tail(s) to interpret probabilities.
• The standard normal table gives left-tail probabilities; for right-tail areas use the complement or symmetry: P(Z > z) = \Phi(-z).
• A common source of confusion in informal notes is mislabeling the median for the standard normal. In a symmetric distribution like the standard normal, the mean is 0 and the median is also 0; the value 0.8577 referenced in the transcript corresponds to the cumulative probability at z ≈ 1.07, since $\Phi(1.07) \approx 0.8577.$

Formulas to remember (LaTeX)

• Transformation: $Z = \frac{X - \mu}{\sigma}, \quad Z \sim N(0,1).$
• Original to standard normal probability: $P(X \le x) = \Phi\left( \frac{x - \mu}{\sigma} \right).$
• Standard normal CDF: $\Phi(z) = P(Z \le z).$
• Right-tail: P(Z > z) = 1 - \Phi(z) = \Phi(-z).
• Symmetry relation: $\Phi(z) = 1 - \Phi(-z).$
• Example values: $\Phi(1.28) \approx 0.8997, \quad 1 - \Phi(1.28) \approx 0.1003.$