Discrete Random Variables and Probability Distributions

3-1 Discrete Random Variables

• Discrete random variables can take on a countable number of values.
• Example: The number of lines in use in a business communication system can range from 0 to 48.

3-2 Probability Distributions and Probability Mass Functions

• Probability mass function (PMF) for a discrete random variable X is defined as:
  • Conditions:
  1. f(x) \geq 0
  2. \sum f(x) = 1
  3. f(x) = P(X = x_i)
• Example: If a wafer has a 0.01 probability of contamination, the PMF can represent sequences of outcomes.

3-3 Cumulative Distribution Functions

• The cumulative distribution function (CDF) F(x) is defined as:
  • F(x) = P(X \leq x) = \sum{xi \leq x} f(x_i)
• Properties of CDF:
  1. 0 \leq F(x) \leq 1
  2. If x < y , then F(x) \leq F(y) .

3-4 Mean and Variance of a Discrete Random Variable

• Mean (Expected Value):
  • \mu = E(X) = \sum x f(x)
• Variance:
  • \sigma^2 = \sum (x - \mu)^2 f(x) = \sum x^2 f(x) - \mu^2
• Standard Deviation:
  • \sigma = \sqrt{\sigma^2}

3-5 Discrete Uniform Distribution

• Definition: A random variable X has a discrete uniform distribution if each value in its range has equal probability:
  • f(x) = \frac{1}{n}
• Mean of X:
  • \mu = E(X) = \frac{a + b}{2}
• Variance of X:
  • \sigma^2 = \frac{(b - a + 1)^2 - 1}{12}

3-6 Binomial Distribution

• Definition: A binomial random variable arises from n independent Bernoulli trials, where each trial has two possible outcomes.
• Probability mass function:
  • P(X = x) = {n \choose x} p^x (1 - p)^{n - x}
• Expected value and variance:
  • Mean: \mu = np
  • Variance: \sigma^2 = np(1 - p)

3-9 Poisson Distribution

• Definition: A random variable X follows a Poisson distribution if counts occur randomly in fixed intervals.
• Probability Mass Function:
  • f(x) = \frac{e^{-\lambda} \lambda^x}{x!} , where \lambda is the average number of events in the interval.
• Expected value and variance for Poisson:
  • Mean: E(X) = \lambda
  • Variance: V(X) = \lambda

Application Examples

• Example 1: Probability distribution of a communication system with 48 lines.
• Example 2: Probability of contamination in semiconductor wafers follows a geometric distribution.
• Example 3: Analyzing production parts for defects using a cumulative distribution function.