October 13

Normal (Gaussian) Distribution

A continuous probability distribution defined by its mean ($\mu$) and variance ($\sigma^2$). It is characterized by its symmetric, bell-shaped curve.

• Notation: $X \sim N(\mu, \sigma^2)$
• Parameters:
  • $\mu$: Mean (center of the distribution)
  • $\sigma^2$: Variance (spread of the distribution)
  • $\sigma$: Standard Deviation

• Probability Density Function (PDF):

  \[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]

The Standard Normal Distribution

A special case of the normal distribution where the mean is 0 and the standard deviation is 1.

• Notation: $Z \sim N(0, 1)$
• Parameters: $\mu = 0$, $\sigma = 1$

Z-Scores

A z-score measures the number of standard deviations a data point ($x$) is from the mean ($\mu$). It’s used to standardize any normal distribution, allowing us to use a single table (the Z-table) to find probabilities.

• Formula:

  \[Z = \frac{X - \mu}{\sigma}\]

• Interpretation:

  • A positive z-score means the value is above the mean.
  • A negative z-score means the value is below the mean.

Finding Probabilities using Z-Scores

Probabilities for a normal distribution correspond to the area under the curve. To find this area:

1. Calculate the z-score for the given value(s) of $X$.
2. Look up the z-score in a standard normal (Z) table or use a calculator to find the cumulative probability, $P(Z \le z)$.

• $P(X \le a) = P\left(Z \le \frac{a - \mu}{\sigma}\right)$
• $P(X > a) = 1 - P(X \le a)$
• $P(a \le X \le b) = P(X \le b) - P(X \le a)$

The Empirical Rule (68-95-99.7)

A useful approximation for the area under the curve for values falling within 1, 2, or 3 standard deviations of the mean.

• ~68% of data lies within $1\sigma$ of the mean ($\mu \pm \sigma$).
• ~95% of data lies within $2\sigma$ of the mean ($\mu \pm 2\sigma$).
• ~99.7% of data lies within $3\sigma$ of the mean ($\mu \pm 3\sigma$).

---

Examples

Problem: Test scores are normally distributed with a mean of 80 and a standard deviation of 5. So, $X \sim N(80, 5^2)$.

1. What is the probability a student scores 75 or less?
   • Calculate z-score: $Z = \frac{75 - 80}{5} = -1.0$
   • Find probability: $P(X \le 75) = P(Z \le -1.0)$
   • From a Z-table, $P(Z \le -1.0) \approx 0.1587$.
   • Answer: There’s a 15.87% chance.
2. What is the probability a student scores between 82 and 90?
   • Calculate z-scores:
     • For $x=82$: $Z_1 = \frac{82 - 80}{5} = 0.4$
     • For $x=90$: $Z_2 = \frac{90 - 80}{5} = 2.0$
   • Find probabilities from Z-table:
     • $P(Z \le 2.0) \approx 0.9772$
     • $P(Z \le 0.4) \approx 0.6554$
   • Calculate the difference: $P(82 \le X \le 90) = P(Z \le 2.0) - P(Z \le 0.4) = 0.9772 - 0.6554 = 0.3218$.
   • Answer: There’s a 32.18% chance.