Topic 1: Formula Reference
Key formulas and definitions for discrete probability distributions.
schoolDefinitions
chevron_rightRandom VariableBasics
A function that assigns real-valued numbers to each possible outcome of a random experiment.
chevron_rightDiscrete Random VariableBasics
A random variable that can assume a countable number of values (e.g., 0, 1, 2, ...).
chevron_rightContinuous Random VariableBasics
A random variable that can assume an uncountable number of values (e.g., any real number in an interval).
chevron_rightSample SpaceBasics
The set of all possible outcomes of a random experiment.
Discrete PMF
Probability Mass Function
The probability that discrete random variable Y takes value y.
PMF Axiom 1
Every probability must be between 0 and 1 (inclusive).
PMF Axiom 2
All probabilities must sum to exactly 1.
Discrete CDF
Cumulative Distribution Function
The probability that X is less than or equal to x. For discrete variables, this is a step function.
PMF from CDF
For integer-valued discrete RVs, the PMF is the jump height in the CDF.
Expectation
Expected Value (Mean)
The weighted average of all possible values, weighted by their probabilities.
Expectation of a Function
Expected value of any function of X.
Variance
Variance Definition
Measures the spread of the distribution around the mean.
Variance (Computational Formula)
Alternative formula that is often easier to compute.
Expectation Properties
Expectation of Linear Combination
Expectation is a linear operator.
Variance Properties
Variance of Linear Combination
Adding a constant does not change variance; scaling by a multiplies variance by a².
MGF
Moment Generating Function
A function that generates moments when differentiated.
First Moment from MGF
The first derivative of MGF at t=0 gives the mean.
Second Moment from MGF
The second derivative of MGF at t=0 gives E(X²).
Special Distributions
Hypergeometric PMF
Sampling without replacement from a finite population.
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Use the computational formula V(X) = E(X²) - [E(X)]² for faster variance calculations!