6.1: The Standard Normal Distribution

Discrete Random Variables:
- A discrete random variable takes on specific, distinct values with gaps between permissible outcomes (e.g., variables restricted strictly to whole numbers where fractions and decimals are excluded).
- Probability as Area: The probability of selecting a specific outcome equals the area of its corresponding rectangle in a histogram. The probability of selecting a value from a set of outcomes equals the sum of the areas of those rectangles.
- Binomial Distribution Example: A binomial distribution $B(n, p)$ is a discrete random variable. As the sample size $n$ becomes large, its histogram forms a single symmetric bump centered near the mean ( $\mu = np$ ) and tapers off at both tails.
Continuous Random Variables:
- A continuous random variable can take on any real value within a continuous interval. Outcomes are not restricted to whole numbers; fractions, decimals, and every real number in between are possible outputs.
Measuring Spread via Standard Deviation:
- The standard deviation of any dataset serves as a measuring stick indicating how far a data value lies from the mean.
- Standardizing a raw score $x$ is performed using the z-score formula: $z = \frac{x - \mu}{\sigma}$

Notation: A normal random variable is denoted as $X \sim N(\mu, \sigma)$ .
Probability Density Function:
- The probability density function defining the normal distribution curve is: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$
Parameters:
- The normal distribution depends on two parameters:
- Mean ( $\mu$ ): Specifies the center and location of the peak.
- Standard Deviation ( $\sigma$ ): Determines how spread out the distribution is and controls the speed at which the curve tapers off.
Graphical Characteristics:
- Symmetry: The graph is perfectly symmetric about the mean ( $\mu$ ), containing zero skewness.
- Peak: The peak occurs at the center where $x = \mu$ , representing the highest point on the curve.
- Tapering: The curve tapers off smoothly in both directions away from the mean towards the horizontal axis.
- Transformation Properties:
- Changing $\mu$ shifts the graph horizontally along the horizontal axis.
- Changing $\sigma$ stretches or squishes the curve vertically and horizontally.

Probabilities as Area Under the Curve:
- For continuous distributions, probabilities correspond directly to the area underneath the distribution curve.
- The cumulative probability of observing a value less than $x$ is written as $P(X < x)$ and corresponds to the area under the normal curve up to $x$
- Total Area Property: The total area underneath the normal distribution curve equals exactly $1$ .

Definition:
- The standard normal distribution is the specific normal distribution where the mean $\mu = 0$ and the standard deviation $\sigma = 1$ , expressed as $N(0, 1)$ .
Simplification of Z-Score in Standard Normal:
- For the standard normal distribution $N(0, 1)$ , substituting $\mu = 0$ and $\sigma = 1$ simplifies the calculation: $z = \frac{x - \mu}{\sigma} = \frac{x - 0}{1} = x$
Purpose of Z-Scores:
- The z-score standardizes data values from any arbitrary normal distribution $N(\mu, \sigma)$ , allowing comparison to the standard normal distribution $N(0, 1)$ .
- A z-score answers the specific question: "What data value in the standard normal distribution corresponds to this data value from a given normal distribution?"

Question: What can be concluded about a data value from a normal distribution if its z-score is negative? What if its z-score is positive?
Response:
- Negative Z-Score ( $z < 0$ ): Indicates that the data value is strictly smaller than the mean ( $x < \mu$ ) and lies to the left of the center on the distribution curve.
- Positive Z-Score ( $z > 0$ ): Indicates that the data value is strictly greater than the mean ( $x > \mu$ ) and lies to the right of the center on the distribution curve.
- Zero Z-Score ( $z = 0$ ): Indicates that the data value is exactly equal to the mean ( $x = \mu$ ).

Calculating Z-Scores from Data Values:
- Example 1:
- Given $X \sim N(5.8, 2)$ and $x = 7$ : $z = \frac{7 - 5.8}{2} = \frac{1.2}{2} = 0.6$
- Example 2:
- Given $Y \sim N(18, 1.4)$ and $y = 15$ : $z = \frac{15 - 18}{1.4} = \frac{-3}{1.4} \approx -2.143$
- Example 3:
- Given $P \sim N(3.25, 0.82)$ and $p = 4.1$ : $z = \frac{4.1 - 3.25}{0.82} = \frac{0.85}{0.82} \approx 1.037$
Calculating Data Values from Z-Scores:
- Given $X \sim N(80, 5.7)$ , find the corresponding value $x$ using $x = \mu + z\sigma$ :
- Case 1 ( $z = -2.3$ ): $x = 80 + (-2.3)(5.7) = 80 - 13.11 = 66.89$
- Case 2 ( $z = 0.1$ ): $x = 80 + (0.1)(5.7) = 80 + 0.57 = 80.57$
- Case 3 ( $z = 1.6$ ): $x = 80 + (1.6)(5.7) = 80 + 9.12 = 89.12$

For any normal distribution $X \sim N(\mu, \sigma)$ , data distribution across standard deviations follows the empirical rule:
- Within 1 Standard Deviation: Approximately $68\%$ of the data falls within 1 standard deviation of the mean, lying in the interval $[\mu - 1\sigma, \mu + 1\sigma]$ .
- Within 2 Standard Deviations: Approximately $95\%$ of the data falls within 2 standard deviations of the mean, lying in the interval $[\mu - 2\sigma, \mu + 2\sigma]$ .
- Within 3 Standard Deviations: Approximately $99\%$ of the data falls within 3 standard deviations of the mean, lying in the interval $[\mu - 3\sigma, \mu + 3\sigma]$ .