🃏Engineering Probability Unit 4 – Discrete Random Variables & Distributions

Key Concepts and Definitions

• Discrete random variables take on a countable number of distinct values
• Sample space is the set of all possible outcomes of a random experiment
• Events are subsets of the sample space
• Probability is a measure of the likelihood of an event occurring
• Random variables assign numerical values to outcomes in a sample space
• Discrete random variables have a probability distribution that specifies the probability of each possible value
• Independence means the occurrence of one event does not affect the probability of another event

Types of Discrete Random Variables

• Bernoulli random variables have only two possible outcomes (success or failure)
• Binomial random variables count the number of successes in a fixed number of independent Bernoulli trials
  • Characterized by the number of trials $n$ and the probability of success $p$
• Geometric random variables count the number of trials until the first success occurs
• Poisson random variables count the number of events occurring in a fixed interval of time or space
  • Characterized by the average rate of occurrence $\lambda$
• Hypergeometric random variables count the number of successes in a fixed number of draws without replacement from a finite population

Probability Mass Functions (PMF)

• PMF denoted as $P(X = x)$ gives the probability that a discrete random variable $X$ takes on a specific value $x$
• PMF satisfies two conditions:
  • $P(X = x) \geq 0$ 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, graph, or formula
• PMF allows calculation of probabilities for specific values or ranges of values
• 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 denoted as $F(x) = P(X \leq x)$ gives the probability that a random variable $X$ takes on a value less than or equal to $x$
• CDF is a non-decreasing function with values between 0 and 1
• CDF can be obtained by summing the PMF values up to and including $x$
  • $F(x) = \sum_{t \leq x} P(X = t)$
• CDF allows calculation of probabilities for intervals and ranges of values
• Example: For a Bernoulli random variable with $p = 0.7$, the CDF is $F(0) = 0.3$ and $F(1) = 1$

Expected Value and Variance

• Expected value (mean) of a discrete random variable $X$ is the weighted average of all possible values
  • $E(X) = \sum_{x} x \cdot P(X = x)$
• Variance measures the spread or dispersion of a random variable around its expected value
  • $Var(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2$
• Standard deviation is the square root of the variance
• Expected value and variance provide important summary statistics for a random variable
• Linearity of expectation: $E(aX + bY) = aE(X) + bE(Y)$ for constants $a$ and $b$

Common Discrete Distributions

• Bernoulli distribution: Models a single trial with two possible outcomes (success with probability $p$, failure with probability $1-p$)
• Binomial distribution: Models the number of successes in a fixed number of independent Bernoulli trials
  • PMF: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ for $k = 0, 1, ..., n$
• Geometric distribution: Models the number of trials until the first success occurs
  • PMF: $P(X = k) = (1-p)^{k-1} p$ for $k = 1, 2, ...$
• Poisson distribution: Models the number of events occurring in a fixed interval of time or space
  • PMF: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$ for $k = 0, 1, 2, ...$
• Hypergeometric distribution: Models the number of successes in a fixed number of draws without replacement from a finite population
  • PMF: $P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$ for $\max(0, n-(N-K)) \leq k \leq \min(n, K)$

Properties and Applications

• Discrete random variables have a countable number of possible values
• Discrete distributions are characterized by their PMF, CDF, expected value, and variance
• Discrete distributions are used to model various real-world phenomena:
  • Bernoulli: Success/failure of a single trial (coin flip, defective item)
  • Binomial: Number of successes in a fixed number of trials (defective items in a batch, successful free throws)
  • Geometric: Number of trials until the first success (number of attempts to win a game)
  • Poisson: Number of events in a fixed interval (number of customers arriving per hour, number of defects per unit area)
  • Hypergeometric: Number of successes in a fixed number of draws without replacement (defective items in a sample from a lot)
• Discrete distributions can be used to calculate probabilities, make predictions, and inform decision-making

Problem-Solving Techniques

• Identify the type of discrete random variable and its parameters
• Write the PMF or CDF based on the given information
• Use the PMF or CDF to calculate probabilities for specific values or ranges of values
  • $P(X = x)$ for a specific value $x$
  • $P(a \leq X \leq b)$ for an interval $[a, b]$
• Calculate the expected value and variance using the formulas or properties of the specific distribution
• Apply the linearity of expectation to solve problems involving multiple random variables
• Use the Poisson approximation to the binomial distribution when $n$ is large and $p$ is small
• Recognize when to use each type of discrete distribution based on the problem context and assumptions
• Interpret the results in the context of the problem and make appropriate conclusions or decisions