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)≥0f(x) \geq 0
    2. ∑f(x)=1\sum f(x) = 1
    3. f(x)=P(X=xi)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≤x)=∑<em>x</em>i≤xf(xi)F(x) = P(X \leq x) = \sum<em>{x</em>i \leq x} f(x_i)
  • Properties of CDF:
    1. 0≤F(x)≤10 \leq F(x) \leq 1
    2. If x < y , then F(x)≤F(y)F(x) \leq F(y).

3-4 Mean and Variance of a Discrete Random Variable

  • Mean (Expected Value):
    • μ=E(X)=∑xf(x)\mu = E(X) = \sum x f(x)
  • Variance:
    • σ2=∑(x−μ)2f(x)=∑x2f(x)−μ2\sigma^2 = \sum (x - \mu)^2 f(x) = \sum x^2 f(x) - \mu^2
  • Standard Deviation:
    • σ=σ2\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)=1nf(x) = \frac{1}{n}
  • Mean of X:
    • μ=E(X)=a+b2\mu = E(X) = \frac{a + b}{2}
  • Variance of X:
    • σ2=(b−a+1)2−112\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)=(nx)px(1−p)n−xP(X = x) = {n \choose x} p^x (1 - p)^{n - x}
  • Expected value and variance:
    • Mean: μ=np\mu = np
    • Variance: σ2=np(1−p)\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)=e−λλxx!f(x) = \frac{e^{-\lambda} \lambda^x}{x!}, where λ\lambda is the average number of events in the interval.
  • Expected value and variance for Poisson:
    • Mean: E(X)=λE(X) = \lambda
    • Variance: V(X)=λV(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.