🎲Intro to Statistics Unit 4 – Discrete Random Variables

What Are Discrete Random Variables?

• Discrete random variables are variables that can only take on a countable number of distinct values
• Unlike continuous random variables, discrete random variables have a finite or countably infinite number of possible outcomes
• Examples of discrete random variables include the number of heads in a series of coin flips or the number of defective items in a batch of products
• Discrete random variables are often denoted by uppercase letters (X, Y, Z) and their specific values by lowercase letters (x, y, z)
• The probability of a discrete random variable taking on a specific value is described by a probability mass function (PMF)
• Discrete random variables are commonly used in scenarios involving counting, such as the number of successes in a fixed number of trials or the number of events occurring in a given time interval
• The sum of probabilities for all possible values of a discrete random variable equals 1

Probability Mass Functions (PMF)

• A probability mass function (PMF) is a function that describes the probability distribution of a discrete random variable
• The PMF assigns a probability to each possible value of the discrete random variable
• For a discrete random variable X, the PMF is denoted as P(X = x), where x is a specific value that X can take
• The PMF satisfies two conditions:
  • P(X = x) ≥ 0 for all values of x
  • The sum of P(X = x) over all possible values of x equals 1
• The PMF can be represented as a table, graph, or formula, depending on the nature of the discrete random variable
• The cumulative distribution function (CDF) of a discrete random variable is the sum of the PMF values up to a given point
• The CDF, denoted as F(x), represents the probability that the random variable X takes on a value less than or equal to x

Expected Value and Variance

• The expected value (or mean) of a discrete random variable is a measure of the central tendency of its probability distribution
• For a discrete random variable X with PMF P(X = x), the expected value is calculated as: $E(X) = \sum_{x} x \cdot P(X = x)$
• The expected value represents the average value of the random variable over a large number of trials
• The variance of a discrete random variable measures the spread or dispersion of its probability distribution around the expected value
• For a discrete random variable X with PMF P(X = x), the variance is calculated as: $Var(X) = E(X^2) - [E(X)]^2$
  • $E(X^2)$ is the expected value of the squared random variable, calculated as: $E(X^2) = \sum_{x} x^2 \cdot P(X = x)$
• The standard deviation is the square root of the variance and provides a measure of the average distance between the random variable's values and its expected value

Common Discrete Distributions

• Bernoulli distribution: Models a single trial with two possible outcomes (success or failure), with a fixed probability of success (p)
• Binomial distribution: Models the number of successes in a fixed number of independent Bernoulli trials, with a constant probability of success (p) for each trial
• Poisson distribution: Models the number of events occurring in a fixed interval of time or space, given an average rate of occurrence (λ)
• Geometric distribution: Models the number of trials needed to achieve the first success in a series of independent Bernoulli trials, with a constant probability of success (p) for each trial
• Hypergeometric distribution: Models the number of successes in a fixed number of draws from a population without replacement, where the population consists of a known number of successes and failures
• Negative binomial distribution: Models the number of failures before a specified number of successes is achieved in a series of independent Bernoulli trials, with a constant probability of success (p) for each trial

Calculating Probabilities

• To calculate probabilities for discrete random variables, use the probability mass function (PMF) specific to the distribution
• For the binomial distribution with parameters n (number of trials) and p (probability of success), the PMF is given by: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
  • $\binom{n}{k}$ represents the binomial coefficient, which can be calculated as: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• For the Poisson distribution with parameter λ (average rate of occurrence), the PMF is given by: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$
• To find the probability of a range of values, sum the PMF values for each value in the range
• For cumulative probabilities, use the cumulative distribution function (CDF) or sum the PMF values up to the desired value
• When working with tables or graphs, be careful to identify the correct probability values and use the appropriate formulas or methods for the given distribution

Applications in Real Life

• Quality control: The binomial distribution can be used to model the number of defective items in a batch of products, helping manufacturers make decisions about product inspection and acceptance
• Call centers: The Poisson distribution can be used to model the number of calls arriving at a call center within a given time interval, aiding in staffing and resource allocation decisions
• Insurance claims: The negative binomial distribution can be used to model the number of claims filed by policyholders, assisting insurance companies in setting premiums and managing risk
• Inventory management: The geometric distribution can be used to model the number of items sold before restocking is needed, helping businesses optimize their inventory levels and minimize costs
• Clinical trials: The hypergeometric distribution can be used to model the number of patients responding to a treatment when drawing a sample from a population with a known number of responders and non-responders
• Rare events: The Poisson distribution is often used to model the occurrence of rare events, such as the number of earthquakes in a given region or the number of traffic accidents at a particular intersection

Key Formulas and Concepts

• Discrete random variable: A variable that can only take on a countable number of distinct values
• Probability mass function (PMF): A function that describes the probability distribution of a discrete random variable, denoted as P(X = x)
• Expected value: The average value of a discrete random variable over a large number of trials, calculated as $E(X) = \sum_{x} x \cdot P(X = x)$
• Variance: A measure of the spread or dispersion of a discrete random variable's probability distribution around its expected value, calculated as $Var(X) = E(X^2) - [E(X)]^2$
• Binomial distribution: Models the number of successes in a fixed number of independent Bernoulli trials, with PMF $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
• Poisson distribution: Models the number of events occurring in a fixed interval of time or space, with PMF $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$
• Cumulative distribution function (CDF): The sum of the PMF values up to a given point, denoted as F(x)

Practice Problems and Examples

1. A fair coin is tossed 5 times. Let X be the number of heads observed. Find the PMF of X and calculate the expected value and variance of X.
2. A car manufacturer has found that 2% of their products are defective. If a batch of 100 cars is selected, find the probability that exactly 3 cars are defective, using the binomial distribution.
3. A call center receives an average of 10 calls per hour. Find the probability that the call center receives exactly 5 calls in a 30-minute period, using the Poisson distribution.
4. A basketball player has a free throw success rate of 80%. Calculate the probability that the player makes at least 3 out of 5 free throws, using the binomial distribution.
5. A company has 20 employees, 5 of whom are managers. If a committee of 4 employees is randomly selected, find the probability that the committee includes exactly 2 managers, using the hypergeometric distribution.
6. A machine produces bolts, and the probability of a bolt being defective is 0.1. Calculate the expected number of bolts that need to be produced until the first defective bolt is encountered, using the geometric distribution.
7. A store has 100 light bulbs in stock, 20 of which are known to be defective. If a customer buys 10 light bulbs at random, find the probability that at most 2 of the purchased bulbs are defective, using the hypergeometric distribution.
8. A factory produces 10,000 items per day, and the probability of an item being defective is 0.001. Use the Poisson distribution to approximate the probability that exactly 5 defective items are produced in a day.