Chapter 6: Continuous Probability Distributions – Study Notes for Business Statistics

Continuous Probability Distributions

Introduction to Probability Distributions

Probability distributions are fundamental tools in statistics for describing how the values of a random variable are distributed. They are classified into two main types: discrete and continuous probability distributions.

• Discrete Probability Distributions: Concerned with random variables that take on countable values (e.g., number of heads in coin tosses).

• Continuous Probability Distributions: Concerned with random variables that can take on any value within a given interval (e.g., time, distance, weight).

Continuous Random Variables

Continuous random variables are outcomes that can assume any numerical value within an interval, typically measured rather than counted.

• Definition: A continuous random variable is a variable that can take on an infinite number of possible values within a specified range.

• Examples: Time spent on a phone call, distance traveled, weight of an object.

• Probability for Specific Values: The probability of a continuous random variable taking on any exact value is theoretically zero; probabilities are assigned to intervals.

• Probability Representation: Probability is represented by the area under the probability distribution curve over an interval.

Types of Continuous Probability Distributions

Continuous probability distributions can take various forms. The most common types discussed in business statistics are:

• Normal Distribution

• Exponential Distribution

Normal Probability Distribution

Characteristics of the Normal Distribution

The normal distribution is a bell-shaped, symmetric probability distribution that is widely used in statistics.

• Shape: Bell-shaped and symmetric around the mean.

• Mean and Median: Both are equal and located at the center of the distribution.

• Area Under the Curve: The total area under the curve is always 1.0. The area to the left and right of the mean is each 0.5.

• Extends Indefinitely: The tails of the distribution extend infinitely in both directions.

Parameters of the Normal Distribution

• Mean ($\mu$): Determines the center of the distribution.

• Standard Deviation ($\sigma$): Determines the spread or width of the distribution.

Normal Probability Density Function

The mathematical expression for the normal probability density function is:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• x = value of the random variable

• $\mu$ = mean

• $\sigma$ = standard deviation

Standardizing with Z-Scores

To compute probabilities, values are often converted to z-scores:

$z = \frac{x - \mu}{\sigma}$

• Z-score at the mean: $z = 0$

• Negative z-scores: Values less than the mean

• Positive z-scores: Values greater than the mean

Standard Normal Distribution

The standard normal distribution is a special case where $\mu = 0$ and $\sigma = 1$.

Calculating Probabilities Using Z-Tables

Probabilities for intervals are found using the standard normal probability table (z-table) or Excel functions.

• Example: If $z = 0.67$, then $P(z < 0.67) = 0.7486$ (area to the left of $z$).

• Area to the right: $P(z > 0.67) = 1 - 0.7486 = 0.2514$

Empirical Rule (68-95-99.7 Rule)

The empirical rule describes the percentage of data within certain standard deviations of the mean in a normal distribution:

• About 68% within 1 standard deviation ($\mu \pm \sigma$)

• About 95% within 2 standard deviations ($\mu \pm 2\sigma$)

• About 99.7% within 3 standard deviations ($\mu \pm 3\sigma$)

Applications and Examples

• Example 1: Time spent on a phone call is normally distributed with $\mu = 12$ minutes, $\sigma = 3$ minutes. Probability that a call lasts 14 minutes or less: Find $z = \frac{14-12}{3} = 0.67$, then use z-table.

• Example 2: Annual snowfall in Minneapolis is normally distributed with $\mu = 46$ inches, $\sigma = 15$ inches. Probability between 30 and 70 inches: Compute $z$ for both values, find corresponding probabilities, and subtract.

Normal Approximation to the Binomial Distribution

When to Use Normal Approximation

The normal distribution can approximate the binomial distribution when the sample size is large and both $np$ and $n(1-p)$ are greater than 5.

• $n$ = sample size

• $p$ = probability of success

• $q = 1-p$ = probability of failure

Continuity Correction

When using the normal approximation for discrete binomial data, a continuity correction of 0.5 is applied to the interval.

• Example: For $P(X 7)$, use $P(X 6.5)$ in the normal distribution.

Excel Functions for Binomial and Normal Probabilities

• BINOM.DIST: =BINOM.DIST(x, n, p, cumulative)

• NORM.DIST: =NORM.DIST(x, mean, standard_dev, cumulative)

Exponential Probability Distribution

Characteristics of the Exponential Distribution

The exponential distribution is a continuous distribution commonly used to model the time between events.

• Parameter: Only one parameter, $\lambda$ (rate or mean number of occurrences).

• Shape: Right-skewed, not symmetric.

• Values: Only non-negative values (e.g., time cannot be negative).

Exponential Probability Density Function

$f(x) = \lambda e^{-\lambda x}$

• $x$ = value of the random variable (time, etc.)

• $\lambda$ = rate parameter (mean number of occurrences per interval)

Exponential Cumulative Distribution Function

$P(x < a) = 1 - e^{-\lambda a}$

• Example: If the average time between customers is 4 minutes ($\lambda = 0.25$ per minute), probability that the next customer arrives within 2 minutes: $P(x < 2) = 1 - e^{-0.25 \times 2} = 0.3935$

Standard Deviation of the Exponential Distribution

$\sigma = \frac{1}{\lambda}$

Excel Functions for Exponential Probabilities

• EXPON.DIST: =EXPON.DIST(x, lambda, cumulative)

• cumulative = TRUE: Returns cumulative probability $P(X < x)$

• cumulative = FALSE: Returns probability density at $x$

Summary Table: Key Properties of Distributions

Distribution	Parameters	Shape	Typical Application
Normal	Mean ($\mu$), Std. Dev. ($\sigma$)	Bell-shaped, symmetric	Measurement data (e.g., heights, weights)
Exponential	Rate ($\lambda$)	Right-skewed	Time between events (e.g., arrivals, failures)
Binomial	Sample size ($n$), Probability ($p$)	Discrete, can be approximated by normal	Number of successes in $n$ trials

Key Takeaways

• Continuous probability distributions are essential for modeling measurement data in business statistics.

• The normal and exponential distributions are the most commonly used continuous distributions.

• Probabilities for continuous variables are calculated over intervals, not at specific points.

• Excel functions and z-tables are practical tools for calculating probabilities.