Discrete Random Variables and Probability Distributions

Discrete Random Variables

Definition and Introduction

A discrete random variable is a function that maps the sample space of a probability experiment to a subset of the natural numbers (or integers). Formally, if X is a discrete random variable, then:

• Notation: X : S → N, where S is the sample space and N is the set of natural numbers.

• Discrete random variables take on countable values, often representing counts or outcomes of experiments.

Example: Coin Tosses

Consider a random variable W that gives the number of tails minus the number of heads in three tosses of a coin. The sample space S consists of all possible outcomes of three coin tosses. Each outcome is mapped to a value w by W.

• Sample Space S: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

• Mapping: For each outcome, calculate the number of tails minus the number of heads.

Probability Distributions of Discrete Random Variables

Probability Mass Function (PMF)

Every discrete random variable has an associated probability distribution. The probability mass function (pmf) gives the probability that the random variable takes a specific value:

• Notation: P(X = x) or p(x)

• Properties:

  • 0 ≤ P(X = xi) ≤ 1 for all i

  • ∑i P(X = xi) = 1

Example: Defective Television Sets

A shipment of 7 television sets contains 2 defective sets. A hotel randomly purchases 4 sets. Let X be the number of defective sets purchased. The probability distribution of X can be found using combinatorial methods and represented as a probability histogram.

• Possible values of X: 0, 1, 2

• Probability calculation: Use the hypergeometric formula to find P(X = x) for each x.

Example: Cars with Side Airbags

If a car agency sells 50% of its inventory with side airbags, the probability distribution for the number of cars with side airbags among the next 4 cars sold is given by the binomial distribution:

• Formula: $P(X = k) = {4 \choose k} (0.5)^k (0.5)^{4-k}$

• Where: k = 0, 1, 2, 3, 4

Common Discrete Probability Distributions

Several important discrete probability distributions are frequently used in statistics:

• Binomial: $P(X = k) = {n \choose k} p^k (1-p)^{n-k}$

• Negative Binomial: $P(X = k) = {n+k-1 \choose k} p^n (1-p)^k$

• Poisson: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

• Geometric: $P(X = k) = p (1-p)^k$

• Hypergeometric: $P(X = k) = \frac{{B \choose k} {G \choose n-k}}{{B+G \choose n}}$

Additional info: Each distribution models different types of random processes, such as fixed numbers of trials (binomial), waiting times (geometric), or rare events (Poisson).

Cumulative Distribution Function (CDF)

Definition and Properties

The cumulative distribution function (CDF) of a discrete random variable X is defined as:

• Formula: $F(x) = P(X \leq x)$

• Properties:

  • 0 ≤ F(x) for all x

  • F(x) is a non-decreasing function

  • $\lim_{x \to \infty} F(x) = 1$

Example: Calculating and Graphing CDF and PMF

Suppose X is a discrete random variable with pmf: P(X = 1) = 0.4, P(X = 2) = 0.2, P(X = 3) = 0.3, P(X = 4) = 0.1.

• Calculate F(3): $F(3) = P(X \leq 3) = 0.4 + 0.2 + 0.3 = 0.9$

• Graph: The CDF and PMF can be plotted to visualize the distribution.

Example: Sum of Two Dice

Roll two fair 6-sided dice. Let X be the sum of the two dice. The probability distribution table for X lists all possible sums (2 to 12) and their probabilities. To find F(4), sum the probabilities for X = 2, 3, and 4.

• Probability Distribution Table: Each sum has a specific probability based on the number of combinations that yield that sum.

• F(4): $F(4) = P(X \leq 4)$

Example: Imperfections in Fabric

The probability distribution of X, the number of imperfections per 10 meters of synthetic fabric, is given by:

x	0	1	2	3	4
f(x)	0.38	0.36	0.16	0.07	0.03

• CDF Construction: The CDF at each x is the sum of all probabilities up to and including x.

Example: Cumulative Probability for Cars with Airbags

For the car agency example, the cumulative probability distribution for the number of cars with side airbags among the next 4 cars sold is:

• Formula: $F(k) = P(X \leq k) = \sum_{i=0}^{k} {4 \choose i} (0.5)^i (0.5)^{4-i}$

Survival Function

Definition

The survival function S(x) is defined as the probability that the random variable X exceeds x:

• Formula: $S(x) = 1 - F(x) = P(X > x)$

• The survival function is commonly used in reliability theory and survival analysis.