Continuous and Standard Normal Probability Distributions

Continuous Probability Distributions Overview

• Mathematical Model: Describes the likelihood of outcomes occurring across a continuous range of values rather than discrete points.

• Point Probability: The probability that a continuous random variable assumes any exact single value is always zero ($P(X = x) = 0$) due to the infinite number of possible outcomes ($\frac{1}{\infty} = 0$).

• Interval Probability: Probabilities are determined exclusively over an interval ($x_1 \le X \le x_2$) and correspond directly to the area under the probability density function ($f(x)$) curve:

$P(x_1 \le X \le x_2) = \int_{x_1}^{x_2} f(x)\,dx$

• Total Area: The total area under the probability density curve across all possibilities equals $1$.

• Common Continuous Distributions: Uniform (constant horizontal density across an interval), normal (bell-shaped), exponential, Weibo, logistics, and Nakagami.

Properties of the Normal Distribution

• Historical Origin: Developed by the French mathematician Abraham de Moura around 1733.

• Shape and Symmetry: Bell-shaped curve that is perfectly symmetrical around the central mean ($\mu$). The area on either side of the mean is exactly $0.5$.

• Measures of Central Tendency: The peak of the curve occurs where the mean, median, and mode are all equal ($\text{Mean} = \text{Median} = \text{Mode} = \mu$).

• Asymptotic Tails: The tails extend infinitely in both directions, approaching but never touching the horizontal axis.

• Parameters:

  • Mean ($\mu$): Determines the horizontal location/shift of the center (can be negative, zero, or positive).

  • Standard Deviation ($\sigma$): Determines the spread and height. A smaller $\sigma$ produces a steep, narrow peak; a larger $\sigma$ flattens and spreads the distribution.

• Empirical Rule:

  • $\mu \pm 1\sigma$ contains approximately $68.3\%$ of all values.

  • $\mu \pm 2\sigma$ contains approximately $95.4\%$ of all values.

  • $\mu \pm 3\sigma$ contains approximately $99.7\%$ of all values.

The Standard Normal Distribution and Z-Scores

• Definition: A standardized normal distribution with fixed parameters where the mean is zero ($\mu = 0$) and the standard deviation is one ($\sigma = 1$).

• Z-Score Formula: Converts any raw score ($X$) from a normal distribution into standard deviation units:

$Z = \frac{X - \mu}{\sigma}$

• Interpretation: The $Z$-score indicates the number of standard deviations an observation lies above (positive) or below (negative) the distribution mean.

Probability Calculations Using the Z-Table

• Cumulative Principle: Standard cumulative $Z$-tables provide values representing the area from negative infinity up to a specific limit ($P(Z \le z)$).

• Standard Left-Tail Form: Look up $z$ directly in the table to obtain $P(Z \le z)$.

• Upper-Tail Probabilities: Use the complement rule since total area equals $1$:

$P(Z \ge z) = 1 - P(Z \le z)$

• Interval Probabilities: Subtract the cumulative lower limit from the cumulative upper limit:

$P(a \le Z \le b) = P(Z \le b) - P(Z \le a)$

• Equality Notation: Because point probabilities equal zero, boundary inequalities are mathematically equivalent:

$P(Z \le z) = P(Z < z)$

Inverse Calculations (Finding Values from Probabilities)

• Finding $k$ in the Standard Normal Domain:

  • Express the condition in cumulative form ($P(Z \le k)$).

  • Locate the target probability in the body of the $Z$-table.

  • Read off the corresponding row and column values to determine the $Z$-score ($k$).

• Converting $Z$ Back to Raw Values ($X$):

  • Rearrange the transformation formula to solve for $X$:

$X = \mu + Z \times \sigma$

• Example (Edgar's Clothing Store): With $\mu = 244$ and $\sigma = 68$, identifying the threshold for the highest-spending $15\%$ ($P(X \ge k) = 0.15$) corresponds to a cumulative lower probability of $0.85$. Finding the closest table probability ($0.8508$) yields $Z = 1.04$, giving a minimum transaction value of:

$k = 244 + (1.04 \times 68) = 314.72$

Microsoft Excel Functions for Normal Distributions

• Standard Normal Functions ($Z$-domain):

  • NORM.S.DIST(z, TRUE): Computes the cumulative probability $P(Z \le z)$ for a given $Z$-score.

  • NORM.S.INV(probability): Computes the standard $Z$-value corresponding to a given cumulative probability.

• General Normal Functions ($X$-domain):

  • NORM.DIST(x, mean, standard_dev, TRUE): Computes the cumulative distribution function ($\text{CDF}$) probability $P(X \le x)$.

  • NORM.INV(probability, mean, standard_dev): Computes the variable value $x$ corresponding to a specified cumulative probability.