Chapter Overview
This chapter introduces discrete random variables and their distributions. You will learn how to describe the probability behavior of a discrete random variable using its probability mass function (PMF) and cumulative distribution function (CDF), and study important families like the hypergeometric, binomial, and Poisson distributions.
6
Sections
8
Simulators
80
Practice Problems
~20%
Of Exam P
Key Formulas You'll Learn
PMF Properties
f(x) > 0, Σ f(x) = 1
CDF Definition
F(x) = P(X ≤ x)
Hypergeometric
f(x) = C(N₁,x)C(N₂,n-x)/C(N,n)
Expected Value
E[u(X)] = Σ u(x)f(x)
Hypergeometric Mean
μ = n(N₁/N)
Geometric Mean
μ = 1/p
Variance Shortcut
σ² = E(X²) - μ²
MGF Definition
M(t) = E(etX)
Geometric Variance
σ² = q/p²
Binomial PMF
f(x) = C(n,x)pxqn-x
Binomial Mean/Var
μ = np, σ² = npq
Geometric PMF
f(x) = p·qx-1
Geometric Mean/Var
μ = 1/p, σ² = q/p²
Poisson PMF
f(x) = λxe-λ/x!
Poisson Mean = Var
μ = σ² = λ
Sections
2.1 Random Variables of the Discrete Type
Discrete random variables, PMF, CDF, and the hypergeometric distribution
2.2 Mathematical Expectation
Expected value, mean, linearity of expectation, and distribution means
2.3 Special Mathematical Expectations
Variance, standard deviation, moments, and moment-generating functions
2.4 Binomial Distribution
Bernoulli trials, binomial distribution, and its properties
2.5 Negative Binomial Distribution
Geometric distribution, negative binomial, and waiting for successes
2.6 Poisson Distribution
Poisson process and the Poisson distribution