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FRM Exam Part I · Random Variables

Covariance, Correlation and Joint Distributions for FRM Part I

Updated 11 October 2026 · Fact-checked

A joint distribution gives the probability of two variables taking values together. Sum across it to get marginals, divide by a marginal to get conditionals. Covariance measures how the two move together; correlation is covariance ÷ (σX × σY), bounded between -1 and +1. Use Var(X+Y) = Var X + Var Y + 2Cov(X,Y).

Understand Covariance, Correlation and Joint Distributions

A single random variable has one distribution. Risk questions usually involve two or more, such as returns on two assets. The joint distribution tells you the probability of each pair of outcomes, written P(X = x, Y = y).

From the joint distribution you can recover each variable on its own. The marginal distribution of X is found by adding the joint probabilities over all values of Y: P(X = x) = Σ P(X = x, Y = y). In a joint table, marginals are the row and column totals. The conditional distribution answers "what is the distribution of Y if I know X = x?" You take that row of the table and divide by its total: P(Y = y | X = x) = P(X = x, Y = y) ÷ P(X = x).

X and Y are independent if knowing one tells you nothing about the other. Formally, P(X = x, Y = y) = P(X = x) × P(Y = y) for every pair. A single cell that fails the test is enough to show dependence. Independence also means every conditional distribution equals the marginal.

Covariance measures the direction of co-movement: Cov(X,Y) = E[XY] − E[X]E[Y]. Positive means they tend to move the same way, negative means opposite. Its size depends on units, so it is hard to interpret. Correlation rescales it by both standard deviations, giving a unit-free number between -1 and +1.

Independent variables have zero covariance and zero correlation. The reverse is not true. Correlation captures only linear dependence, so two variables can be strongly dependent and still have zero correlation. This is a favourite exam trap.

Key formulas to remember

Marginal probability
P(X = x) = Σy P(X = x, Y = y)
Sum the joint probabilities across the other variable. Row or column totals in a table.
Conditional probability
P(Y = y | X = x) = P(X = x, Y = y) ÷ P(X = x)
Defined only when P(X = x) > 0.
Independence
P(X = x, Y = y) = P(X = x) × P(Y = y) for all x, y
Must hold in every cell. Equivalent to P(Y | X) = P(Y).
Covariance
Cov(X,Y) = E[(X − μX)(Y − μY)] = E[XY] − E[X]E[Y]
Cov(X,X) = Var(X). Units are the product of the units of X and Y.
Correlation
ρ = Cov(X,Y) ÷ (σX × σY)
Always between -1 and +1. Measures linear dependence only.
Variance of a sum
Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X,Y)
For a difference, the covariance term has the opposite sign when a = b = 1: Var(X − Y) = Var X + Var Y − 2Cov.
Covariance scaling
Cov(aX + b, cY + d) = ac Cov(X,Y)
Constants added do not change covariance. Correlation is unchanged by positive scaling.
Expected value of a sum
E(aX + bY) = aE(X) + bE(Y)
Holds whether or not X and Y are independent.

How to solve Covariance, Correlation and Joint Distributions questions

Use this order for any joint distribution question, whether it gives a table or summary statistics.

  1. 1Identify what is given: a joint probability table, or variances and a covariance or correlation.
  2. 2If you have a table, check that the probabilities sum to 1 and compute the marginals from row and column totals.
  3. 3For a conditional probability, pick the relevant row or column and divide each cell by its total.
  4. 4To test independence, compare one cell with the product of its two marginals. One mismatch means dependent.
  5. 5Compute E[X], E[Y] and E[XY] = Σ x·y·P(x,y), then Cov = E[XY] − E[X]E[Y].
  6. 6Convert to correlation by dividing by σX × σY, where σ = √(E[X²] − E[X]²).
  7. 7For sums or portfolios, apply Var(aX + bY) with the covariance term, then take the square root if volatility is asked.
  8. 8Sanity check: correlation must lie in [-1, +1] and variance cannot be negative.

Quickest way: Table shortcut with totals

When to use it: When the question gives a small joint table and asks for a marginal, conditional or independence check.

  1. Write the row and column totals in the margins first. They answer marginal questions at once.
  2. For a conditional, divide the single cell by the total of the given condition, not by 1.
  3. For independence, test the cell with the clearest numbers. Failure ends the question.
  4. For covariance, compute only E[XY] over nonzero-product cells. Zero values of X or Y drop out.
  5. In sum-variance questions, skip the table and plug into Var X + Var Y + 2Cov directly.

Common mistakes in Covariance, Correlation and Joint Distributions

  • Dividing by the wrong total for a conditional probability

    Students divide by the grand total of 1 or by the wrong variable's marginal.

    Fix: Divide by the marginal of the event you are conditioning on, the one after the "|" sign.

  • Concluding independence from zero correlation

    Independence implies zero correlation, and the reverse feels natural.

    Fix: Zero correlation only rules out linear dependence. Check the product rule in the cells; for example, Y = X² can have zero correlation with X yet be fully dependent.

  • Forgetting the 2 in the covariance term

    Students recall Var X + Var Y + Cov from memory.

    Fix: Remember it as expanding (a + b)²: the cross term is 2ab Cov(X,Y).

  • Using the wrong sign for Var(X − Y)

    Students assume variances subtract.

    Fix: Variances never subtract. Var(X − Y) = Var X + Var Y − 2Cov(X,Y). Positive correlation reduces the variance of a difference.

  • Mixing variance and standard deviation in correlation

    The formula needs σ, but problems often give variance.

    Fix: Take square roots first. Correlation = Cov ÷ (σX σY), not Cov ÷ (VarX VarY).

  • Treating a correlation outside [-1, 1] as plausible

    Arithmetic slips go unchecked.

    Fix: Always check the bound. If you get 1.4, recheck E[XY] and the standard deviations.

Worked examples

Example 1

X takes values 0 and 1, Y takes values 0 and 1. Joint probabilities: P(0,0) = 0.30, P(0,1) = 0.20, P(1,0) = 0.10, P(1,1) = 0.40. Find P(Y = 1 | X = 1) and Cov(X,Y).

Show the solution
  1. Marginal of X: P(X = 0) = 0.30 + 0.20 = 0.50; P(X = 1) = 0.10 + 0.40 = 0.50.
  2. Marginal of Y: P(Y = 1) = 0.20 + 0.40 = 0.60.
  3. Conditional: P(Y = 1 | X = 1) = 0.40 ÷ 0.50 = 0.80.
  4. E[X] = 0.50; E[Y] = 0.60.
  5. E[XY] = 1 × 1 × 0.40 = 0.40 (all other cells have a zero product).
  6. Cov = 0.40 − 0.50 × 0.60 = 0.40 − 0.30 = 0.10.

Answer: P(Y = 1 | X = 1) = 0.80 and Cov(X,Y) = 0.10. Since 0.80 ≠ 0.60, X and Y are not independent.

Example 2

Two assets have return volatilities of 10% and 20%, with a correlation of 0.25. A portfolio holds 60% in asset A and 40% in asset B. Find the portfolio volatility.

Show the solution
  1. Cov = ρ σA σB = 0.25 × 0.10 × 0.20 = 0.005.
  2. Var = (0.6)²(0.10)² + (0.4)²(0.20)² + 2(0.6)(0.4)(0.005).
  3. (0.36)(0.01) = 0.0036.
  4. (0.16)(0.04) = 0.0064.
  5. 2 × 0.6 × 0.4 × 0.005 = 0.0024.
  6. Var = 0.0036 + 0.0064 + 0.0024 = 0.0124.
  7. Volatility = √0.0124 ≈ 0.1114.

Answer: Portfolio volatility ≈ 11.14%.

Exam tips

  • Questions often ask for a conditional probability from a table. Write the marginals first to avoid dividing by the wrong total.
  • Expect conceptual items: "zero correlation implies independence?" The answer is no, except in special cases such as jointly normal variables.
  • For portfolio variance, work in decimals and square root only at the end.
  • Remember that adding a constant to a variable leaves covariance unchanged, while multiplying by a constant scales it.
  • A financial calculator's two-variable statistics mode can give covariance from data, but for small tables hand calculation is faster.

Practice questions from Random Variables

Covariance, Correlation and Joint Distributions in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Covariance, Correlation and Joint Distributions: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction of co-movement but its size depends on the units. Correlation divides covariance by both standard deviations, so it is unit-free and lies between -1 and +1. That makes correlation easier to compare across pairs.

How do I find a marginal distribution from a joint distribution?

Add the joint probabilities over all values of the other variable. For a table, the row totals give the marginal of the row variable and the column totals give the marginal of the column variable.

What is the formula for the variance of the sum of two random variables?

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y). If the variables are independent, covariance is zero and the variances simply add. For a difference, the covariance term is subtracted.

Does zero covariance mean the variables are independent?

No. Independence implies zero covariance, but zero covariance only rules out a linear relationship. Nonlinear dependence can still exist.