CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

Discrete and Continuous Random Variables for CFA Level I

Q: What is the difference between a PMF and a PDF?

A PMF applies to discrete variables and gives the actual probability of each value. A PDF applies to continuous variables and gives a density, so probability comes from the area under the curve over an interval. PMF values sum to 1, and the PDF's total area is 1.

Q: Why is the probability of a single value zero for a continuous variable?

A continuous variable has infinitely many possible values in any interval. A single point has no width, so its area under the PDF is zero. Only ranges have positive probability.

Q: What does the CDF tell me in finance?

It gives the probability that a variable, such as a return, is at or below a given level. This lets you find the chance of a loss or a shortfall, and the probability between two levels by subtraction.

Q: Is a stock return discrete or continuous?

Returns are usually modelled as continuous because they can take many values on a range. Prices are quoted in fixed ticks, so they are technically discrete, but the continuous model is a useful approximation.

Updated 7 October 2026 · Fact-checked

A random variable takes numerical values determined by chance. A discrete one has countable outcomes and uses a probability mass function, where P(X = x) is a real probability. A continuous one takes any value in a range and uses a density function, where only areas give probability. The CDF gives P(X ≤ x) for both.

Understand Discrete and Continuous Random Variables

A random variable is a quantity whose value is uncertain until it is observed. Examples: the number of bond defaults in a portfolio this year, or tomorrow's return on a stock. A probability distribution tells you the possible values and how likely each is.

A discrete random variable has a countable set of outcomes, such as 0, 1, 2, 3 defaults. Its probability mass function (PMF) gives p(x) = P(X = x) for each value. Each p(x) is a true probability, so it lies between 0 and 1, and all p(x) add up to 1.

A continuous random variable can take any value in an interval, such as a return of 2.37% or 2.371%. There are infinitely many values, so the probability of any single exact value is 0. The probability density function (PDF), f(x), describes how concentrated probability is near x. Probability comes from the area under the curve between two points. The total area is 1. The height f(x) is not a probability and can exceed 1.

The cumulative distribution function (CDF), F(x) = P(X ≤ x), works for both types. For a discrete variable you add up PMF values up to x, so the CDF is a step function. For a continuous variable it is the area under the PDF up to x, so it is a smooth curve. The CDF never decreases, starts near 0 and ends at 1.

In practice, price quotes and counts are discrete. Returns are usually modelled as continuous because it makes the maths simpler. For a continuous variable, P(a ≤ X ≤ b) = F(b) − F(a), and it makes no difference whether the endpoints are included.

Key formulas to remember

PMF conditions (discrete): 0 ≤ p(x) ≤ 1 for every x; Σ p(x) = 1; p(x) = P(X = x). Use this to find a missing probability in a table.
PDF conditions (continuous): f(x) ≥ 0; total area under f(x) = 1; f(x) is a density, not a probability. It can be greater than 1.
CDF definition: F(x) = P(X ≤ x); Non-decreasing, between 0 and 1. Discrete: F(x) = Σ p(xi) for xi ≤ x.
Interval probability (continuous): P(a ≤ X ≤ b) = F(b) − F(a); For a continuous variable P(X = a) = 0, so endpoints do not matter.
Interval probability (discrete): P(a < X ≤ b) = F(b) − F(a); For discrete variables, check whether the lower endpoint is included. If it is, use F(b) − F(a) + p(a).
Upper tail: P(X > x) = 1 − F(x); Fast way to get 'greater than' probabilities.

How to solve Discrete and Continuous Random Variables questions

Use this routine for any question on PMF, PDF or CDF.

1Decide whether the variable is discrete (countable values) or continuous (any value in a range).
2Write down what is asked in terms of X: P(X = x), P(X ≤ x), P(X > x) or P(a < X ≤ b).
3If continuous and the question asks for P(X = x), the answer is 0. Stop there.
4If discrete, list the values that satisfy the condition and add their PMF probabilities.
5If you are given a CDF, use differences: P(a < X ≤ b) = F(b) − F(a), and 1 − F(x) for upper tails.
6If a probability is missing, use the rule that all probabilities sum to 1 (or total area is 1).
7Check that the answer is between 0 and 1 and that you treated endpoints correctly.

Quickest way: Three-second classification and elimination

When to use it: Use when the question is conceptual or gives a small table and you have about 90 seconds.

Ask: can I list the outcomes one by one? Yes means discrete, no means continuous.
Any option claiming a single exact value of a continuous variable has positive probability is wrong.
Any option saying a PDF height is a probability is wrong.
For a CDF table, subtract rows. For a PMF table, add rows.
Eliminate options that are below 0, above 1, or that make the total differ from 1.

Common mistakes in Discrete and Continuous Random Variables

Treating the PDF height f(x) as a probability.
The PMF height is a probability, so students assume the PDF works the same way.
Fix: Remember that for a continuous variable only area is probability. The height can exceed 1.
Saying P(X = x) is positive for a continuous variable.
It feels odd that a possible value has zero probability.
Fix: There are infinitely many values, so each single point has probability 0. Only intervals have positive probability.
Using F(b) − F(a) for a discrete variable without checking endpoints.
Students carry over the continuous rule, where endpoints do not matter.
Fix: F(b) − F(a) equals P(a < X ≤ b). If the question includes a, add p(a).
Reading the CDF as P(X = x) or P(X ≥ x).
The letter F is mistaken for the PMF or the upper tail.
Fix: F(x) always means P(X ≤ x). For 'at least', use the complement.
Calling the number of trades or defaults continuous because the numbers are large.
Size is confused with countability.
Fix: If outcomes can be counted in whole units, the variable is discrete, however large it is.

Worked examples

Example 1

The number of credit downgrades X in a portfolio next quarter has this PMF: P(X=0) = 0.30, P(X=1) = 0.35, P(X=2) = 0.20, P(X=3) = 0.10, P(X=4) = k. What is P(X ≥ 2)? A) 0.30 B) 0.35 C) 0.40

Show the solution

The probabilities must sum to 1: 0.30 + 0.35 + 0.20 + 0.10 + k = 1.
0.95 + k = 1, so k = 0.05.
P(X ≥ 2) = P(2) + P(3) + P(4) = 0.20 + 0.10 + 0.05 = 0.35.
Check with the complement: 1 − (0.30 + 0.35) = 0.35.

Answer: B) 0.35

Example 2

A continuous random variable, the monthly return X on a fund, has CDF values F(−2%) = 0.10, F(0%) = 0.35 and F(3%) = 0.80. What is P(0% < X ≤ 3%)? A) 0.35 B) 0.45 C) 0.80

Show the solution

For a CDF, P(a < X ≤ b) = F(b) − F(a).
P(0% < X ≤ 3%) = F(3%) − F(0%) = 0.80 − 0.35 = 0.45.
Option A is F(0%), which is P(X ≤ 0%), not the interval probability.
Option C is F(3%), which is P(X ≤ 3%) and wrongly includes all returns up to 0%.

Answer: B) 0.45

Exam tips

Expect conceptual items that test whether a PDF value can be a probability. It cannot.
With a CDF table, subtract rows for intervals and use 1 − F(x) for 'greater than'.
In a PMF table, solve for the missing probability first, then answer.
Watch the wording 'at most', 'less than' and 'at least'. For discrete variables they change which values count.
Do not spend calculator time here. Most items need only addition or subtraction.

Practice questions from Statistical Distributions for Financial Asset Prices and Returns

Discrete and Continuous Random Variables: frequently asked questions

What is the difference between a PMF and a PDF?

A PMF applies to discrete variables and gives the actual probability of each value. A PDF applies to continuous variables and gives a density, so probability comes from the area under the curve over an interval. PMF values sum to 1, and the PDF's total area is 1.

Why is the probability of a single value zero for a continuous variable?

A continuous variable has infinitely many possible values in any interval. A single point has no width, so its area under the PDF is zero. Only ranges have positive probability.

What does the CDF tell me in finance?

It gives the probability that a variable, such as a return, is at or below a given level. This lets you find the chance of a loss or a shortfall, and the probability between two levels by subtraction.

Is a stock return discrete or continuous?

Returns are usually modelled as continuous because they can take many values on a range. Prices are quoted in fixed ticks, so they are technically discrete, but the continuous model is a useful approximation.