Probability Distributions
D is the distribution; z ∼ D describes sampling from it.
From Distribution to Sample
Probability notation becomes easier to read when you begin with D. The symbol D names a distribution over a set. It describes how values are selected from that set. The expression z ∼ D then connects a particular variable, z, with that distribution: z is sampled according to D.
The symbol z does not identify one permanently fixed value. It represents a value obtained through the sampling process described by D. Once this relationship is established, later expressions can use z to evaluate a function, an event, or an expected value.
Reading the Sampling Statement
Read z ∼ D as: z is sampled according to D. Start at D, which identifies the distribution. Then read z as the variable receiving a value from that distribution. The statement describes a sampling relationship, not a permanent assignment to one fixed value.
Generated example: Suppose D describes how a value is selected from a set of colors. Writing z ∼ D means that z is treated as the color produced by a draw governed by D. The notation does not say that z must always be one particular color; it establishes how z is obtained.
Functions and Events
After z ∼ D has established the sampling context, the notation inside brackets determines what is being summarized. Expected-value notation summarizes a real-valued function of the sampled variable. The expression E with subscript z ∼ D followed by f(z) denotes the expected value of f(z) when z is sampled according to D.
Probability notation instead asks how often a condition is true. If f maps Z to true or false, then P with subscript z ∼ D followed by f(z) denotes the probability that f(z) is true when z is sampled according to D. In set language, this is the probability assigned by D to the values for which the condition is true.
| Notation type | What it summarizes | Question it answers |
|---|---|---|
| Expected value | A real-valued function of a sampled variable | What expected quantity is associated with f(z) under D? |
| Probability of an event | Whether a Boolean condition is true | How likely is f(z) to be true under D? |
Separating a Quantity from a Condition
Let z be sampled according to D. Compare an expression that uses f(z) as a real-valued quantity with one that uses f(z) as a true-or-false condition.
Identify the sampled variable: The phrase z ∼ D says that z is sampled according to D.
Read the expected-value expression: An expected value summarizes the real-valued function f(z) over samples governed by D.
Read the probability expression: A probability expression summarizes whether the condition f(z) is true over samples governed by D.
Expected-value notation summarizes a value produced by a function, while probability notation summarizes the truth of an event.
Independent Tuples
D describes individual samples from Z. When m points are sampled from D, the result is an m-component tuple written as (z₁, …, zₘ). The notation D^m refers to the probability over Z^m induced by this process, where each zᵢ is sampled from D independently of the other points.
The superscript m does not describe one individual sample raised to a numerical power. It signals a distribution over a larger space: the space of m-component sample tuples. The important change is the object being described. D governs individual values, while D^m governs complete tuples produced by repeated independent sampling.
Generated example: If m points are sampled from D, one outcome is a tuple such as (z₁, …, zₘ). D^m describes the probability over all such m-component tuples, rather than describing only z₁ or only another single component.
Notation Mistakes
Treating z ∼ D as if it assigned one permanently fixed value to z.
The notation describes z as a value obtained by sampling from D.
Fix:
Read it as: z is sampled according to D.Confusing an expected value with an event probability.
Expected-value notation summarizes a real-valued function, while probability notation summarizes whether a Boolean condition is true.
Fix:
First ask whether the expression summarizes a quantity or the truth of an event.Reading D^m as one individual sample raised to a power.
D^m is the induced probability over m-component tuples in Z^m.
Fix:
Read the superscript as indicating repeated independent sampling and a tuple-valued outcome.Ignoring the distribution subscript when reading an expectation or probability.
The subscript identifies the distribution governing the variable inside the brackets.
Fix:
Use the subscript to identify which sampling process the expression summarizes.
Check Your Reading
For each statement, identify what is being described: an individual sampled value, an expected quantity, an event probability, or a distribution over tuples. Then explain the role of D or D^m.
Hints
- Begin with D: it names the distribution over the underlying set.
- For z ∼ D, focus on how z is obtained.
- For an expected value, look for a real-valued function.
- For an event probability, look for a Boolean condition.
- For D^m, look for m independently sampled components forming a tuple.
Key Takeaways
- D names a probability distribution over a set.
- z ∼ D means that z is sampled according to D.
- Expected-value notation summarizes a real-valued function of a sampled variable.
- Probability notation summarizes when a Boolean condition is true.
- D^m describes the induced probability over m-component tuples formed by independent samples from D.
Key Takeaways
- D is the distribution governing individual samples from a set.
- The statement z ∼ D establishes that z is sampled according to D.
- Expected values summarize real-valued quantities, whereas event probabilities summarize true-or-false conditions.
- D^m is a distribution over m-component tuples in Z^m created by independent sampling from D.