Continuous Probability Distributions

• Continuous Probability Distributions are explored.

Learning Objectives

• Continuous random variables should be described
• Probabilities for continuous uniform distribution variables can be calculated and interpreted
• Normal distributions and standard normal distributions should be described
• Probabilities can to be calculated and interpreted for normal distribution variables
• Calculate and interpret probabilities for variables with exponential distribution
• Calculate and interpret probabilities for variables with lognormal distribution

Introductory Case: Demand for Salmon

• Akiko Hamaguchi, the manager at Little Ginza sushi restaurant in Phoenix, Arizona, needs to estimate daily salmon
• Daily salmon consumption is normally distributed, with a mean of 12 pounds and a standard deviation of 3.2 pounds
• Buying 20 pounds of salmon daily results in excessive wastage
• Akiko needs to find the probability that the demand for salmon at Little Ginza is above 20 pounds
• Akiko needs to find the probability that the demand for salmon at Little Ginza is below 15 pounds
• Akiko needs to determine the amount of salmon to buy daily, satisfying 90% of the demand

Continuous Random Variables and the Uniform Distribution

• A continuous random variable is characterized by uncountable values in an interval
• The values cant be described with a list
• Return on a mutual fund or the time to complete a task are examples
• The probability that a continuous random variable assumes a particular value is zero, unlike a discrete random variable
• Nonzero probability cant be assigned to each of the uncountable values and the probabilities sum to one
• Probability needs to be be calculated a specific interval

Probability Density Function

• Graph approximates the relative frequency polygon for the population, for all possible values of X
• The area under all values of X must equal one
• The probability the variable assumes a value with an interval is defined as the area under, between points a and b
• For any value x of a continuous random variable X, the cumulative probability is a possibility

Normal Distribution

• The normal probability distribution is a bell-shaped distribution, also known as Gaussian distribution
• It is the most extensively used distribution
• It closely approximates the probability distribution for a wide range of random variables
• Analyze the underlying data to determine appropriateness of normal distribution
• Use histograms and boxplots
• It is assumed that random variables are normally distributed here
• The normal distribution is the cornerstone of statistical inference

Characteristics of Normal Distribution

• Normal distribution is bell-shaped and symmetric around its mean; mean/median/mode are the same
• It is described by two parameters: the mean and the variance
• It's asymptotic as tails get closer to the horizontal axis without touching it
• A graph depicting the normal probability density function is known as the normal curve or bell curve
• Use the cumulative distribution function to compute probabilities: the area under the normal curve up to the value
• Tables or software should be used to find probabilities

Examples of Normal Distribution

• Ages of employees in industries A, B, and C are examples
• If the mean age of employees in industry A is greater than Industry B, the normal curve for Industry A is to the right of Industry B
• If the standard deviation for Industry A is less than Industry C, the normal curve for Industry A is less dispersed, with a higher peak

Standard Normal Distribution

• The standard normal distribution is a special case of the normal distribution, denoted by Z
• The mean is zero
• The standard deviation is one
• Lowercase z denotes a value Z may assume
• The value z is the z-score from Chapter 3
• The number of standard deviations a given value is away from the mean can be detemined
• Converting values into z-scores is called standardizing
• Text includes a standard normal table/z table
• Text provides areas under the z curve
• Left-hand page: z values less than or equal to 0
• Right-hand page: z values greater than or equal to 0

Transforming Random Variables

• Any normally distributed random variable can be transformed into the standard normal random variable
• If X has a normal distribution with mean and standard deviation, it can be transformed into z

Applications

• Scores on a management aptitude exam are normally distributed with a mean of 72 and a standard deviation of 8
• The probability can be tested for a randomly selected manager scoring above 60
• The probability can be tested for a randomly selected manager scoring between 68 and 84

Inverse Transformation

• Given probabilities, the inverse transformation can be used to compute values of X
• For example: Scores on a management aptitude exam are normally distributed with a mean of 72 and a standard deviation of 8
• The lowest score to place a manager in the top 10% (90th percentile) of the distribution can be determined
• The highest score to place a manager in the bottom 25% (25th percentile) of the distribution can be determined

Binomial Probabilities

• Computing binomial probabilities for larger values of n is tedious
• With large values of n, the binomial distribution can be approximated by the normal distribution
• Computing binomial probabilities with Excel and R is easy
• Normal distribution approximation is crucial when making inference for the population proportion, p

Excel and R

• Excel and R Functionality can be used.