Chapter 6: Continuous Probability Distributions

Chapter Overview

• This chapter introduces the concept of continuous probability distributions within the context of business statistics.

Learning Objectives (LOs)

• LO 6.1: Describe a continuous random variable.

• LO 6.2: Calculate and interpret probabilities for a random variable that follows the continuous uniform distribution.

• LO 6.3: Calculate and interpret probabilities for a random variable that follows the normal distribution.

• LO 6.4: Calculate and interpret probabilities for a random variable that follows the exponential distribution.

Introductory Case: Demand for Salmon

• Context: Akiko, manager of Little Ginza (a sushi restaurant in Phoenix, Arizona), estimates daily salmon demand.

• Estimation: Daily consumption is normally distributed with:

  • Mean: 12 pounds

  • Standard deviation: 3.2 pounds

• Challenges: Previously, buying 20 pounds daily led to excess waste.

• Objectives for Akiko:

  1. Calculate the probability that demand exceeds 20 pounds.

  2. Calculate the probability that demand is below 15 pounds.

  3. Determine the optimal amount of salmon to buy daily to meet demand 90% of the time.

6.1 Continuous Random Variables and the Uniform Distribution

• Definition: A continuous random variable can take on an infinite number of values within a given range.

  • Characteristics: Uncountable values, not listable.

  • Example: Return on a mutual fund or time taken to complete a task.

• Probability Assignment: Unlike discrete variables, the probability of a continuous random variable assuming a particular value is zero.

  • Probability is calculated within an interval, i.e., $p(a ≤ x ≤ b)$.

Probability Density Function (PDF)

• Counterpart to Probability Mass Function (PMF): The PDF is denoted as $f(x)$ and is characterized by the following:

  • $f(x) ext{ is } ext{nonnegative}$ for all values of $x$.

  • The area under the curve (or graph of $f(x)$) over its entire range equals one.

• Cumulative Distribution Function (CDF): Denoted as $F(x)$, where:

  • $F(x) = p(X ≤ x)$

  • Represents the area under the PDF up to point $x$.

6.1 Continuous Uniform Distribution

• Definition: A continuous uniform distribution describes a scenario where every value in the interval $[a,b]$ is equally likely to be observed.

• Characteristics:

  • PDF for Continuous Uniform Distribution:
    $f(x) = \frac{1}{b-a} \text{ for } a ≤ x ≤ b$

  • Expected Value (Mean):
    $E(X) = \frac{a + b}{2}$

  • Standard Deviation:
    $SD(X) = \frac{b - a}{\sqrt{12}}$

• Graphical Interpretation:

  • Height of the PDF is constant and the probability corresponds to the area under the curve (i.e., a rectangle).

Example: Sales Distribution

• Sales Scenario: Sales for a cosmetic line follow a continuous uniform distribution with:

  • Lower limit: $a = 2,500$

  • Upper limit: $b = 5,000$

  • Calculate Mean and Standard Deviation:

  • $E(X) = \frac{2500 + 5000}{2} = 3750$

  • $SD(X) = \frac{5000 - 2500}{\sqrt{12}} = 720.35$

  • Probability Calculations:

  1. Probability sales exceed $4,000$: Calculate the area between $4,000$ and $5,000$.

  2. Probability sales are between $3,200$ and $3,800$: Find the area between these two values under the PDF.

6.2 The Normal Distribution

• Definition: The normal distribution, also known as the bell-shaped or Gaussian distribution, is crucial in statistics.

• Characteristics:

  • Symmetrical about its mean.

  • Mean, median, and mode are equal.

  • Defined by its population mean $$ and population variance $^2$.

  • Asymptotic: Tails approach the horizontal axis but never touch it.

Calculating Probabilities

• Use of CDF: Use the CDF $F(x)$ to find probabilities as the area under the normal curve up to value $x$.

• Excel Functionality: Utilize Excel to compute probabilities for given values.

Example: Management Aptitude Scores

• Setup: Scores are normally distributed with:

  • Mean: $072$

  • Standard deviation: $08$

• Probability Calculations:

  • Probability of scoring above 60:

  • $P(X > 60) = 1 - NORM.DIST(60, 72, 8, TRUE)$

  • Approx. Probability = $0.9332$

  • Probability of scoring between 68 and 84:

  • $P(68 ≤ X ≤ 84)$ needs computation via Excel.

Standard Normal Distribution

• Definition of Standard Normal Distribution: Denoted by $Z$.

  • Properties: Mean = 0, Standard deviation = 1.

  • Z-score represents the number of standard deviations from the mean.

• Standardization Process: Conversion from $X$ to $Z$ given by:

  • $Z = \frac{X - \mu}{\sigma}$

  • Example scenario allows converting normal distribution problems to standard normal distribution.

6.3 The Exponential Distribution

• Definition: The exponential distribution is a continuous probability distribution that is useful for modeling the time between events in a Poisson process.

  • Related to the Poisson distribution, where:

  • Poisson Distribution: Represents the number of occurrences of an event in a specified interval.

  • Exponential Distribution: Represents the time until the next occurrence.

• Properties of Exponential Random Variable:

  • Nonnegative: Defined for $x ≥ 0$.

  • Parameters: Rate parameter $0B$ indicating the average number of events per time unit.

• Probability Density Function (PDF):

  • $f(x) = 0B e^{-0B x}$ for $x ≥ 0$

  • Mean and standard deviation for exponential random variable are both $0B^{-1}$.

Example: Email Response Times

• Context: The time between emails follows an exponential distribution with a mean of 25 minutes.

• Calculations:

  1. Rate parameter ($0B$): $0B = \frac{1}{25} = 0.04 ext{ emails per minute}$

  2. Probability calculations:

  • Probability of not receiving an email for over an hour:

    • $P(X > 60) = e^{-0.04 imes 60} ≈ 0.0907$

  • Probability of receiving an email within 10 minutes:

    • $P(X ≤ 10) = 1 - e^{-0.04 imes 10} ≈ 0.3297$

• Excel Computations:

  • Use functions similar to previous examples to evaluate probabilities.