Continuous Probability Distributions (Chapter 6): Study Notes for Business Statistics

Continuous Probability Distributions

Introduction

Continuous probability distributions are fundamental in business statistics for modeling and analyzing data that can take on any value within a given interval. This chapter focuses on three major types of continuous probability distributions: normal, exponential, and uniform. Understanding these distributions allows business professionals to make informed decisions based on probabilistic models.

6.1 Continuous Random Variables

Definition and Properties

• Continuous random variables are variables that can assume any numerical value within a specified interval, as determined by an experiment or observation.

• The probability that a continuous random variable takes on any exact value is always zero; probabilities are assigned to intervals, not points.

• The value observed depends on the measurement's level of precision.

Example: The time (in minutes) a customer spends on a service call is a continuous random variable because it can take any value within a range (e.g., 0 to 60 minutes).

6.2 Normal Probability Distributions

Characteristics of the Normal Distribution

• The normal distribution is bell-shaped and symmetrical around its mean.

• The mean and median are equal due to symmetry.

• Most values cluster around the mean, with probabilities tapering off as values move further from the center.

• The total area under the curve is 1.0, representing the entire probability space.

• The curve extends indefinitely in both directions, never touching the horizontal axis.

Parameters and Effects

• The distribution is fully described by its mean ($\mu$) and standard deviation ($\sigma$).

• Changing $\mu$ shifts the distribution left or right.

• Changing $\sigma$ increases or decreases the spread (width) of the distribution.

Probability Density Function (PDF)

The mathematical expression for the normal distribution is:

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

• $e = 2.71828$ (base of natural logarithms)

• $\mu$ = mean

• $\sigma$ = standard deviation

• $x$ = value of interest

Standardization and z-Scores

• The z-score indicates how many standard deviations a value $x$ is from the mean:

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

• z-scores are negative for values below the mean, positive for values above the mean, and zero at the mean.

The Standard Normal Distribution

• A normal distribution with $\mu = 0$ and $\sigma = 1$ is called the standard normal distribution.

• Probabilities for any normal distribution can be found by converting to z-scores and using the standard normal table.

Calculating Probabilities Using Normal Tables

• Standard normal tables provide the cumulative probability to the left of a given z-score.

• To find $P(x \leq a)$, calculate the z-score for $a$ and look up the corresponding area.

• To find $P(x > a)$, subtract the cumulative probability from 1.

Example: If $x = 14$, $\mu = 12$, $\sigma = 3$:

$z = \frac{14 - 12}{3} = 0.67$

From the table, $P(x \leq 14) = 0.7486$, so $P(x > 14) = 1 - 0.7486 = 0.2514$.

Empirical Rule (68-95-99.7 Rule)

• For a normal distribution:

  • About 68% of values fall within 1 standard deviation of the mean.

  • About 95% within 2 standard deviations.

  • About 99.7% within 3 standard deviations.

Finding Probabilities for Intervals

• To find the probability that $x$ falls between two values, calculate the z-scores for both and subtract the smaller cumulative probability from the larger.

Example: Annual snowfall in Minneapolis is normally distributed with $\mu = 46$ in., $\sigma = 18.5$ in. Probability that snowfall is between 30 and 70 inches:

$z_{30} = \frac{30 - 46}{18.5} = -0.86$ $z_{70} = \frac{70 - 46}{18.5} = 1.30$

$P(30 \leq x \leq 70) = P(z \leq 1.30) - P(z \leq -0.86) = 0.9032 - 0.1949 = 0.7083$

6.3 Exponential Probability Distributions

Definition and Applications

• The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process (e.g., time between customer arrivals).

Probability Density Function (PDF)

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

• $\lambda$ = mean number of occurrences per unit interval

• $x$ = value of interest ($x \geq 0$)

Cumulative Distribution Function (CDF)

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

Mean and Standard Deviation

• Mean: $\mu = \frac{1}{\lambda}$

• Standard deviation: $\sigma = \frac{1}{\lambda}$

Example

If the average time between arrivals is 4 minutes, $\lambda = 0.25$ per minute. Probability that the next customer arrives within 2 minutes:

$P(x < 2) = 1 - e^{-0.25 \times 2} = 1 - e^{-0.5} \approx 0.3935$

6.4 Uniform Probability Distributions

Definition and Properties

• The uniform distribution describes a situation where all intervals of the same length within the distribution's range are equally probable.

Probability Density Function (PDF)

$f(x) = \frac{1}{b - a} \quad \text{for} \ a < x < b$ $f(x) = 0 \quad \text{otherwise}$

• $a$ = lower endpoint, $b$ = upper endpoint

Cumulative Distribution Function (CDF)

$P(c < x < d) = \frac{d - c}{b - a}$

Mean and Standard Deviation

• Mean: $\mu = \frac{a + b}{2}$

• Standard deviation: $\sigma = \frac{b - a}{\sqrt{12}}$

Example

If the time to finish an exam is uniformly distributed between 70 and 120 minutes, the probability a student finishes in less than 80 minutes is:

$P(x < 80) = \frac{80 - 70}{120 - 70} = \frac{10}{50} = 0.2$

Mean: $\mu = \frac{70 + 120}{2} = 95$ minutes Standard deviation: $\sigma = \frac{120 - 70}{\sqrt{12}} \approx 14.43$ minutes

Summary Table: Key Properties of Continuous Distributions

Distribution	PDF	Mean ($\mu$)	Std. Dev. ($\sigma$)	Typical Application
Normal	$\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$	$\mu$	$\sigma$	Heights, test scores, measurement errors
Exponential	$\lambda e^{-\lambda x}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda}$	Time between arrivals, service times
Uniform	$\frac{1}{b-a}$ for $a < x < b$	$\frac{a+b}{2}$	$\frac{b-a}{\sqrt{12}}$	Random number generation, equal-likelihood events