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? |
Both expressions use the same sampling context, but they summarize different kinds of information.
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.
- 1Identify the sampled variable
The phrase z ∼ D says that z is sampled according to D.
- 2Read the expected-value expression
An expected value summarizes the real-valued function f(z) over samples governed by D.
- 3Read 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 Dm refers to the probability over Zm 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 Dm 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). Dm describes the probability over all such m-component tuples, rather than describing only z₁ or only another single component.
Notation Mistakes
Common mistake Treating z ∼ D as if it assigned one permanently fixed value to z.
What it looks likeReading the statement as saying that z is always one particular member of Z.
Why it failsThe notation describes z as a value obtained by sampling from D.
Do this insteadRead it as: z is sampled according to D.
Common mistake Confusing an expected value with an event probability.
What it looks likeTreating E with subscript z ∼ D followed by f(z) and P with subscript z ∼ D followed by f(z) as interchangeable.
Why it failsExpected-value notation summarizes a real-valued function, while probability notation summarizes whether a Boolean condition is true.
Do this insteadFirst ask whether the expression summarizes a quantity or the truth of an event.
Common mistake Reading Dm as one individual sample raised to a power.
What it looks likeInterpreting D^m as a modified version of one value sampled from D.
Why it failsDm is the induced probability over m-component tuples in Zm.
Do this insteadRead the superscript as indicating repeated independent sampling and a tuple-valued outcome.
Common mistake Ignoring the distribution subscript when reading an expectation or probability.
What it looks likeTreating the subscript z ∼ D as decorative notation.
Why it failsThe subscript identifies the distribution governing the variable inside the brackets.
Do this insteadUse 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 Dm.
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 Dm, look for m independently sampled components forming a tuple.
Key Takeaways
- 1D names a probability distribution over a set.
- 2z ∼ D means that z is sampled according to D.
- 3Expected-value notation summarizes a real-valued function of a sampled variable.
- 4Probability notation summarizes when a Boolean condition is true.
- 5Dm describes the induced probability over m-component tuples formed by independent samples from D.
At a glance
- 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.
- Dm is a distribution over m-component tuples in Zm created by independent sampling from D.