🎲Intro to Probability Unit 5 – Discrete Random Variables & Distributions

Key Concepts

• Discrete random variables take on a countable number of distinct values
• Probability mass function (PMF) assigns probabilities to each possible value of a discrete random variable
• Cumulative distribution function (CDF) gives the probability that a discrete random variable is less than or equal to a specific value
• Expected value represents the average value of a discrete random variable over many trials
• Variance measures the spread or dispersion of a discrete random variable around its expected value
  • Calculated by taking the average of the squared differences between each value and the expected value
• Independence and mutual exclusivity are important properties of discrete random variables
  • Independent events do not affect each other's probabilities
  • Mutually exclusive events cannot occur simultaneously
• Discrete distributions describe the probabilities of different outcomes for a discrete random variable (coin flips, dice rolls)

Types of Discrete Random Variables

• Bernoulli random variables have only two possible outcomes (success or failure)
  • Examples include coin flips (heads or tails) and yes/no survey questions
• 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 represent the number of trials needed to achieve the first success in a series of independent Bernoulli trials
• Poisson random variables model the number of events occurring in a fixed interval of time or space
  • Characterized by the average rate of occurrence $\lambda$
• Hypergeometric random variables describe the number of successes in a fixed number of draws from a population without replacement
• Negative binomial random variables represent the number of failures before a specified number of successes in a series of independent Bernoulli trials
• Discrete uniform random variables assign equal probabilities to a finite set of values

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
• Denoted as $P(X = x)$, where $X$ is the random variable and $x$ is a possible value
• The sum of all probabilities in a PMF must equal 1
  • $\sum_{x} P(X = x) = 1$
• PMFs can be represented as tables, formulas, or graphs
  • In a PMF graph, the height of each point represents the probability of the corresponding value
• The PMF of a sum of independent discrete random variables is the convolution of their individual PMFs
• PMFs are used to calculate probabilities, expected values, and other properties of discrete random variables

Cumulative Distribution Functions (CDF)

• A cumulative distribution function (CDF) gives the probability that a discrete random variable is less than or equal to a specific value
• Denoted as $F(x) = P(X \leq x)$, where $X$ is the random variable and $x$ is a possible value
• CDFs are non-decreasing functions, meaning that $F(a) \leq F(b)$ if $a \leq b$
• The CDF can be obtained by summing the PMF values for all values less than or equal to $x$
  • $F(x) = \sum_{t \leq x} P(X = t)$
• CDFs can be used to calculate probabilities for intervals of values
  • $P(a < X \leq b) = F(b) - F(a)$
• The CDF of a discrete random variable is a step function, with jumps at each possible value
• CDFs are useful for comparing different discrete distributions and determining percentiles

Expected Value and Variance

• The expected value (or mean) of a discrete random variable is the weighted average of all possible values
  • Denoted as $E(X)$ or $\mu$
  • Calculated by summing the product of each value and its probability: $E(X) = \sum_{x} x \cdot P(X = x)$
• The variance of a discrete random variable measures the average squared deviation from the expected value
  • Denoted as $Var(X)$ or $\sigma^2$
  • Calculated using the formula: $Var(X) = E(X^2) - [E(X)]^2$
• The standard deviation is the square root of the variance and has the same units as the random variable
  • Denoted as $\sigma$
• The expected value and variance have important properties:
  • Linearity of expectation: $E(aX + b) = aE(X) + b$
  • Variance of a constant: $Var(a) = 0$
  • Variance of a sum of independent random variables: $Var(X + Y) = Var(X) + Var(Y)$
• These properties are useful for calculating the expected value and variance of transformed or combined discrete random variables

Common Discrete Distributions

• Binomial distribution: models the number of successes in a fixed number of independent Bernoulli trials
  • PMF: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
  • Expected value: $E(X) = np$
  • Variance: $Var(X) = np(1-p)$
• Poisson distribution: models the number of events occurring in a fixed interval of time or space
  • PMF: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$
  • Expected value: $E(X) = \lambda$
  • Variance: $Var(X) = \lambda$
• Geometric distribution: models the number of trials needed to achieve the first success in a series of independent Bernoulli trials
  • PMF: $P(X = k) = (1-p)^{k-1}p$
  • Expected value: $E(X) = \frac{1}{p}$
  • Variance: $Var(X) = \frac{1-p}{p^2}$
• Hypergeometric distribution: models the number of successes in a fixed number of draws from a population without replacement
  • PMF: $P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$
  • Expected value: $E(X) = n\frac{K}{N}$
  • Variance: $Var(X) = n\frac{K}{N}\left(1-\frac{K}{N}\right)\frac{N-n}{N-1}$
• Negative binomial distribution: models the number of failures before a specified number of successes in a series of independent Bernoulli trials
  • PMF: $P(X = k) = \binom{k+r-1}{k}(1-p)^k p^r$
  • Expected value: $E(X) = \frac{r(1-p)}{p}$
  • Variance: $Var(X) = \frac{r(1-p)}{p^2}$

Properties and Applications

• Discrete random variables have several important properties:
  • The probability of any single value is between 0 and 1 (inclusive)
  • The sum of all probabilities in a PMF equals 1
  • The CDF is a non-decreasing function with values between 0 and 1 (inclusive)
• Discrete distributions can be used to model various real-world phenomena:
  • Binomial distribution: quality control (defective items), medical trials (treatment success)
  • Poisson distribution: call center arrivals, website traffic, radioactive decay
  • Geometric distribution: number of attempts until first success (job interviews, sales)
  • Hypergeometric distribution: sampling without replacement (defective items in a batch)
  • Negative binomial distribution: number of failures before a specified number of successes (customer complaints before a product is redesigned)
• Discrete random variables can be transformed or combined to create new random variables
  • Example: the sum of two independent discrete random variables is a new discrete random variable
• Understanding the properties and applications of discrete random variables is crucial for modeling and analyzing real-world situations

Problem-Solving Techniques

• Identify the type of discrete random variable and its parameters (e.g., binomial with $n$ and $p$)
• Write the PMF or CDF of the random variable using the appropriate formula
• Calculate probabilities using the PMF, CDF, or properties of the distribution
  • For PMF: $P(X = x)$
  • For CDF: $P(a < X \leq b) = F(b) - F(a)$
  • For complement: $P(X > a) = 1 - P(X \leq a)$
• Use the expected value and variance formulas to calculate these properties for the given distribution
• Apply the linearity of expectation and properties of variance when working with transformed or combined random variables
• Recognize when to use the PMF, CDF, or other properties of the distribution to solve a problem
• Interpret the results in the context of the problem, considering the real-world implications
• Verify that the solution makes sense and satisfies any given conditions or constraints