Random Variables: Discrete vs. Continuous

• Anne Jones, a local Starbucks manager, is concerned about competition from trendy coffee shops.
• Loyal customers visit her store an average of 18 times a month.
• Anne needs to decide on staffing needs, balancing costs and customer wait times.
• Understanding customer arrival probability will help Anne calculate expected visits in a 5-day period.
• Plus the likelihood of a customer visiting a specific number of times during that period.

Random Variables Defined

• A random variable assigns numerical values to the outcomes of an experiment, capturing uncertainty.
• It summarizes experiment outcomes with numerical values.
• Letter "X" denotes a random variable.
• Distinct values represent a random variable, for example, the number of employees.

Discrete Random Variables

• Assumes a countable number of distinct values.

Continuous Random Variables

• Characterized by uncountable values in an interval.
• It cannot be summarized with a list, for example, return on a mutual fund.

Probability Distribution

• Every discrete random variable is associated with it
• It provides the probability that a random variable assumes a particular value.
• The probability of each "x" value is between 0 and 1.
• The sum of all probabilities equals 1.
• A discrete random variable can be defined in terms of the cumulative distribution function.
• An example would be a number rolled on a die, considered a discrete uniform, with a finite number of values.
• Each value is equally likely and is symmetric

Expected Value, Variance, and Standard Deviation

• The expected value can also be referred to as the mean.
• The expected value is also a weighted average of all possible values of X.
• It is denoted as and indicates central location.
• The variance and standard deviation measure variability.
• The variance is denoted and calculated as.
• The standard deviation is denoted by and indicates if values are clustered about the mean or widely scattered.
• Risk-averse consumers demand positive expected gain for taking risks and consider declining risky prospects even with potential gains.
• Risk-neutral consumers ignore risk completely and always accept prospects of positive gain.
• Risk-loving consumers may accept risky prospects even if the expected gain is negative.

Binomial Distribution

• Different experiments generate different probability distributions, including binomial, Poisson, and hypergeometric distributions.
• A Bernoulli process comprises a series of "n" independent and identical trials where each trial has only two outcomes: success and failure.
• The probabilities of success and failure remain constant across trials.

Binomial Random Variable "X"

• Number of successes achieved in "n" trials of a Bernoulli process.
• The binomial or binomial probability distribution shows probabilities associated with possible values.
• Examples include customer defaults, consumer reactions to social media campaigns, and drug effectiveness.

Probability

• Probability of "x" successes in "n" Bernoulli trials tells us number of sequences possible with "x" successes and failures
• The second part represents the probability of any particular sequence with "x" successes and failures.