📈Intro to Probability for Business Unit 4 – Discrete Probability Distributions

Key Concepts and Definitions

• Discrete probability distributions assign probabilities to discrete random variables
• Random variables are variables whose values are determined by the outcomes of a random experiment
• Discrete random variables can only take on a finite or countably infinite number of distinct values
• Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1
  • 0 indicates an impossible event, while 1 indicates a certain event
• The sum of probabilities for all possible outcomes in a discrete probability distribution equals 1
• Support of a discrete random variable is the set of all possible values it can take
• Probability distribution functions (PDF) describe the probability of a discrete random variable taking on a specific value

Types of Discrete Probability Distributions

• Bernoulli distribution models a single trial with two possible outcomes (success or failure)
  • Characterized by a single parameter $p$, representing the probability of success
• Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials
  • Characterized by parameters $n$ (number of trials) and $p$ (probability of success in each trial)
• Poisson distribution models the number of events occurring in a fixed interval of time or space
  • Characterized by a single parameter $\lambda$, representing the average number of events per interval
• Geometric distribution models the number of trials until the first success in a series of independent Bernoulli trials
  • Characterized by a single parameter $p$, representing the probability of success in each trial
• Hypergeometric distribution models the number of successes in a fixed number of draws from a finite population without replacement
  • Characterized by parameters $N$ (population size), $K$ (number of successes in the population), and $n$ (number of draws)
• Negative binomial distribution models the number of failures before a specified number of successes in a series of independent Bernoulli trials
  • Characterized by parameters $r$ (number of successes) and $p$ (probability of success in each trial)

Probability Mass Functions (PMF)

• A probability mass function (PMF) is a function that gives the probability of a discrete random variable taking on a specific value
• For a discrete random variable $X$, the PMF is denoted as $P(X = x)$, where $x$ is a possible value of $X$
• The PMF satisfies two conditions:
  1. $P(X = x) \geq 0$ for all $x$ in the support of $X$
  2. $\sum_{x} P(X = x) = 1$, where the sum is taken over all possible values of $X$
• The PMF can be used to calculate probabilities of events involving the discrete random variable
  • $P(a \leq X \leq b) = \sum_{x=a}^{b} P(X = x)$
• The cumulative distribution function (CDF) of a discrete random variable is the sum of its PMF up to a given point
  • $F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t)$

Expected Value and Variance

• The expected value (or mean) of a discrete random variable $X$ is a weighted average of its possible values, weighted by their probabilities
  • $E(X) = \sum_{x} x \cdot P(X = x)$
• The expected value represents the long-run average value of the random variable over many trials
• The variance of a discrete random variable $X$ measures the average squared deviation from its expected value
  • $Var(X) = E((X - E(X))^2) = \sum_{x} (x - E(X))^2 \cdot P(X = x)$
• The standard deviation is the square root of the variance and measures the average deviation from the mean
  • $SD(X) = \sqrt{Var(X)}$
• Linearity of expectation: For any two random variables $X$ and $Y$, $E(X + Y) = E(X) + E(Y)$
• Properties of variance:
  • $Var(aX + b) = a^2 Var(X)$ for constants $a$ and $b$
  • $Var(X + Y) = Var(X) + Var(Y)$ if $X$ and $Y$ are independent

Common Discrete Distributions in Business

• Binomial distribution is used to model the number of successes in a fixed number of trials (e.g., number of defective items in a batch)
• Poisson distribution is used to model the number of rare events occurring in a fixed interval (e.g., number of customer arrivals per hour)
• Geometric distribution is used to model the number of trials until the first success (e.g., number of sales calls until a sale is made)
• Hypergeometric distribution is used to model the number of successes in a fixed number of draws from a finite population without replacement (e.g., number of defective items in a sample drawn from a batch)
• Negative binomial distribution is used to model the number of failures before a specified number of successes (e.g., number of unsuccessful product launches before a successful one)

Practical Applications in Business

• Inventory management: Poisson distribution can model the demand for a product during a fixed time period
• Quality control: Binomial and hypergeometric distributions can model the number of defective items in a batch or sample
• Marketing: Geometric distribution can model the number of sales calls required to make a sale
• Project management: Negative binomial distribution can model the number of unsuccessful projects before a successful one
• Risk assessment: Discrete probability distributions can help quantify the likelihood and impact of various risks
• Decision-making: Expected values and variances can guide decisions by comparing the long-run average outcomes and variability of different options

Calculation Techniques and Examples

• To calculate probabilities using a PMF, substitute the given values into the formula and simplify
  • Example: For a binomial distribution with $n=5$ and $p=0.3$, find $P(X=2)$
    • $P(X=2) = \binom{5}{2} (0.3)^2 (0.7)^3 \approx 0.309$
• To calculate expected values, multiply each possible value by its probability and sum the results
  • Example: For a discrete random variable $X$ with PMF $P(X=1)=0.4$, $P(X=2)=0.3$, and $P(X=3)=0.3$, find $E(X)$
    • $E(X) = 1 \cdot 0.4 + 2 \cdot 0.3 + 3 \cdot 0.3 = 1.9$
• To calculate variances, subtract the expected value from each possible value, square the differences, multiply by the probabilities, and sum the results
  • Example: For the same random variable $X$ as above, find $Var(X)$
    • $Var(X) = (1-1.9)^2 \cdot 0.4 + (2-1.9)^2 \cdot 0.3 + (3-1.9)^2 \cdot 0.3 \approx 0.69$

Key Takeaways and Review

• Discrete probability distributions assign probabilities to discrete random variables, which can only take on a finite or countably infinite number of distinct values
• The sum of probabilities for all possible outcomes in a discrete probability distribution equals 1
• Common discrete probability distributions include Bernoulli, binomial, Poisson, geometric, hypergeometric, and negative binomial distributions
• Probability mass functions (PMF) give the probability of a discrete random variable taking on a specific value and satisfy two conditions: non-negativity and summing to 1
• The expected value is a weighted average of the possible values, while variance measures the average squared deviation from the expected value
• Discrete probability distributions have various applications in business, such as inventory management, quality control, marketing, project management, risk assessment, and decision-making
• To calculate probabilities, expected values, and variances, substitute given values into the appropriate formulas and simplify
• Understanding discrete probability distributions is crucial for making informed decisions and quantifying uncertainty in business contexts