🃏Engineering Probability Unit 10 – Discrete Probability Distributions

Key Concepts and Definitions

• Discrete probability distributions assign probabilities to discrete random variables which can only take on a countable number of distinct values
• Random variables are variables whose values are determined by the outcomes of a random experiment
• Probability is a measure of the likelihood that an event will occur expressed as a number between 0 and 1
  • 0 indicates an impossible event
  • 1 indicates a certain event
• Sample space is the set of all possible outcomes of a random experiment
• Mutually exclusive events cannot occur at the same time (rolling a 1 and a 2 on a die)
• Independent events do not influence each other's probabilities (flipping a coin and rolling a die)
• Conditional probability is the probability of an event occurring given that another event has already occurred

Types of Discrete Probability Distributions

• Bernoulli distribution models a single trial with two possible outcomes (success or failure)
• Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials
• Geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials
• Poisson distribution models the number of events occurring in a fixed interval of time or space given a known average rate
• Hypergeometric distribution models the number of successes in a fixed number of draws without replacement from a finite population
• Negative binomial distribution models the number of failures before a specified number of successes is reached in a series of independent Bernoulli trials
• Discrete uniform distribution assigns equal probabilities to a finite number of outcomes

Probability Mass Functions (PMF)

• PMF is a function that gives the probability of a discrete random variable taking on a specific value
• Denoted as $P(X = x)$ where $X$ is the random variable and $x$ is a specific value
• Properties of a valid PMF:
  • $P(X = x) \geq 0$ for all $x$
  • $\sum_{x} P(X = x) = 1$ (sum of probabilities over all possible values is 1)
• PMF can be represented as a table, formula, or graph
• Helps calculate probabilities of specific events and derive other properties of the distribution
• Example: For a fair six-sided die, the PMF is $P(X = x) = \frac{1}{6}$ for $x = 1, 2, 3, 4, 5, 6$

Cumulative Distribution Functions (CDF)

• CDF gives the probability that a random variable $X$ takes a value less than or equal to $x$
• Denoted as $F(x) = P(X \leq x)$
• Properties of a valid CDF:
  • $0 \leq F(x) \leq 1$ for all $x$
  • $F(x)$ is a non-decreasing function
  • $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$
• Can be derived from the PMF by summing probabilities up to a given value
• Helps calculate probabilities of events involving inequalities
• Example: For a fair six-sided die, the CDF is $F(x) = \frac{x}{6}$ for $x = 1, 2, 3, 4, 5, 6$

Expected Value and Variance

• Expected value (mean) is the average value of a random variable over many trials
  • Denoted as $E(X) = \sum_{x} x \cdot P(X = x)$
  • Represents the center of the distribution
• Variance measures the spread of the distribution around the mean
  • Denoted as $Var(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2$
  • Higher variance indicates more dispersion
• Standard deviation is the square root of the variance
• Helps characterize the distribution and compare different distributions
• Example: For a binomial distribution with $n = 10$ and $p = 0.4$, $E(X) = np = 4$ and $Var(X) = np(1-p) = 2.4$

Common Discrete Distributions in Engineering

• Binomial distribution models the number of defective items in a sample (quality control)
• Poisson distribution models the number of arrivals in a queue (queueing theory)
  • Also models the number of failures in a given time period (reliability engineering)
• Geometric distribution models the number of trials until the first success (reliability testing)
• Hypergeometric distribution models the number of defective items in a sample without replacement (acceptance sampling)
• Negative binomial distribution models the number of failures before a specified number of successes (reliability testing)
• Discrete uniform distribution models random selection from a finite set (Monte Carlo simulations)

Applications in Engineering Problems

• Quality control: Determine the probability of accepting a lot with a certain number of defective items
• Reliability engineering: Calculate the probability of a system failing within a given time period
• Queueing theory: Estimate the probability of a certain number of customers arriving at a service facility
• Acceptance sampling: Decide whether to accept or reject a lot based on the number of defective items in a sample
• Monte Carlo simulations: Generate random samples from a discrete distribution to model a complex system
• Risk analysis: Assess the probability of different outcomes in a project or decision-making process
• Inventory management: Determine the optimal order quantity based on the probability of demand

Solving Practical Examples

• Identify the appropriate discrete probability distribution for the given problem
• Determine the parameters of the distribution based on the given information
• Write the PMF or CDF of the distribution
• Calculate the required probabilities using the PMF, CDF, or properties of the distribution
  • Use the complement rule $P(A) = 1 - P(A^c)$ when necessary
• Interpret the results in the context of the problem
• Example: A machine produces 10% defective items. If 20 items are randomly selected, find the probability of having exactly 3 defective items.
  • Binomial distribution with $n = 20$ and $p = 0.1$
  • $P(X = 3) = \binom{20}{3} (0.1)^3 (0.9)^{17} \approx 0.1901$