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FRM Exam Part I · Fundamentals of Probability

Expected Value and Variance of a Random Variable

Updated 11 October 2026 · Fact-checked

Expected value is the probability-weighted average of a random variable's outcomes: E(X) = Σ x·p(x). Variance is the probability-weighted average squared deviation from the mean: E(X²) − [E(X)]². Standard deviation is the square root of variance and is in the same units as X.

Understand Expectations, Mean and Variance

A random variable assigns a number to each outcome of an uncertain event, such as a bond's one-year payoff or a trading day's profit. To summarise it, you need two things: where it centres and how widely it spreads.

The expected value (mean) measures the centre. You multiply each possible outcome by its probability and add. It is a long-run average, not necessarily a value the variable can take. A fair die has an expected value of 3.5, yet you can never roll 3.5.

The variance measures spread. For each outcome, take the distance from the mean, square it, weight it by its probability and add. Squaring stops positive and negative deviations from cancelling. The cost is that variance is in squared units, such as USD². The standard deviation is the square root of variance. It returns to the original units, so risk managers quote it as volatility.

The key exam skill is the effect of linear transformations. If Y = a + bX, the mean shifts and scales: E(Y) = a + b·E(X). The variance only scales, by b²: Var(Y) = b²·Var(X). Adding a constant moves the whole distribution without changing its spread. The standard deviation scales by |b|.

Expectation is also linear for sums: E(X + Y) = E(X) + E(Y) always, even if X and Y are dependent. Variance of a sum needs covariance, which is covered in related topics.

Key formulas to remember

Expected value (discrete)
E(X) = Σ xᵢ · p(xᵢ)
Probabilities must sum to 1. For a continuous variable, replace the sum with an integral of x·f(x).
Variance (definition)
Var(X) = σ² = E[(X − μ)²] = Σ (xᵢ − μ)² · p(xᵢ)
Use this when the mean is a clean number.
Variance (shortcut)
Var(X) = E(X²) − [E(X)]²
E(X²) = Σ xᵢ² · p(xᵢ). Usually faster with a calculator. Result can never be negative.
Standard deviation
σ = √Var(X)
Same units as X. Not the same as variance.
Linear transformation of the mean
E(a + bX) = a + b·E(X)
Holds for any constants a and b.
Linear transformation of the variance
Var(a + bX) = b² · Var(X)
The constant a drops out. Standard deviation of a + bX is |b| · σ.
Expectation of a sum
E(aX + bY) = a·E(X) + b·E(Y)
Needs no independence assumption.
Expectation of a function
E[g(X)] = Σ g(xᵢ) · p(xᵢ)
In general E[g(X)] ≠ g(E[X]), for example E(X²) ≠ [E(X)]² unless the variance is zero.

How to solve Expectations, Mean and Variance questions

Use this routine for any question on the mean, variance or standard deviation of a random variable.

  1. 1List every outcome xᵢ with its probability p(xᵢ). Check that the probabilities sum to 1; if one is missing, find it as 1 minus the rest.
  2. 2Compute the mean: multiply each xᵢ by p(xᵢ) and add.
  3. 3Compute E(X²): square each xᵢ, multiply by p(xᵢ) and add.
  4. 4Compute variance as E(X²) − μ². Check that it is not negative.
  5. 5Take the square root if the question asks for standard deviation or volatility.
  6. 6If the question gives Y = a + bX, do not recompute from scratch. Apply E(Y) = a + bE(X) and Var(Y) = b²Var(X).
  7. 7Check units and the question wording: variance or standard deviation, and the correct variable (X or Y).

Quickest way: Shortcut with the shortcut formula and transformations

When to use it: Use when outcomes are few and numbers are awkward, or when the question applies a linear transformation to a variable whose moments you already know.

  1. Use Var = E(X²) − μ² instead of squaring deviations. It avoids decimals in the deviations.
  2. On a financial calculator, enter outcomes and probabilities as data points if the model supports weighted statistics; otherwise compute the two sums by hand.
  3. For Y = a + bX, skip the table. Mean: a + b·μ. Standard deviation: |b|·σ.
  4. Before computing, eliminate options that give a negative variance or ignore the b² factor.
  5. If outcomes are symmetric about a value, the mean is that value; this saves the first sum.

Common mistakes in Expectations, Mean and Variance

  • Forgetting to square b when finding Var(a + bX).

    The mean scales by b, so students assume variance does too.

    Fix: Variance is in squared units, so the multiplier is b². Standard deviation uses |b|.

  • Adding the constant a to the variance.

    Students copy the mean rule E(a + bX) = a + bE(X).

    Fix: A constant shift does not change spread. Var(a + bX) has no a term.

  • Using E(X²) = [E(X)]².

    It looks like squaring is harmless inside the expectation.

    Fix: E(X²) exceeds [E(X)]² whenever variance is positive. That gap is exactly the variance.

  • Reporting variance when asked for standard deviation, or the reverse.

    Time pressure and similar-looking options.

    Fix: Underline the requested measure. Finish with a square root if it asks for standard deviation or volatility.

  • Using a negative b and getting a negative standard deviation or variance.

    Applying b instead of b² or |b|.

    Fix: Variance and standard deviation are never negative. Use b² and |b|.

  • Using probabilities that do not sum to 1, or treating a frequency as a probability.

    Skipping the check at the start.

    Fix: Add the probabilities first. If they do not total 1, fix the missing one before calculating.

Worked examples

Example 1

A bond's one-year payoff X (in USD per 100 face value) is 100 with probability 0.90, 60 with probability 0.07, and 0 with probability 0.03. Find the expected payoff and the standard deviation.

Show the solution
  1. Check probabilities: 0.90 + 0.07 + 0.03 = 1.00.
  2. E(X) = 100(0.90) + 60(0.07) + 0(0.03) = 90 + 4.2 + 0 = 94.2.
  3. E(X²) = 100²(0.90) + 60²(0.07) + 0 = 10,000(0.90) + 3,600(0.07) = 9,000 + 252 = 9,252.
  4. Var(X) = 9,252 − 94.2² = 9,252 − 8,873.64 = 378.36.
  5. σ = √378.36 ≈ 19.45.

Answer: Expected payoff = 94.2; standard deviation ≈ 19.45.

Example 2

A trading desk's daily profit X (in USD thousands) has a mean of 40 and a standard deviation of 15. Fees and a fixed charge turn it into net profit Y = 0.8X − 10. Find E(Y) and the standard deviation of Y.

Show the solution
  1. E(Y) = a + b·E(X) = −10 + 0.8(40) = −10 + 32 = 22.
  2. Var(X) = 15² = 225.
  3. Var(Y) = b²·Var(X) = 0.8² × 225 = 0.64 × 225 = 144.
  4. σ_Y = √144 = 12. Check: |b|·σ_X = 0.8 × 15 = 12.

Answer: E(Y) = USD 22 thousand; standard deviation of Y = USD 12 thousand.

Exam tips

  • Questions often hide a linear transformation in a story about fees, leverage or currency conversion. Identify a and b first.
  • Check whether the question asks for variance or standard deviation before you start, and finish with the right one.
  • When two options differ only by a factor of b versus b², you are being tested on the variance rule.
  • Use E(X²) − μ² for tables of outcomes; it is quicker and less error-prone under time pressure.
  • Do not assume independence for the expectation of a sum. You only need it when working with variance of a sum.

Practice questions from Fundamentals of Probability

Expectations, Mean and Variance in other exams

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

Expectations, Mean and Variance: frequently asked questions

What is the difference between variance and standard deviation?

Variance is the average squared deviation from the mean, so it is in squared units. Standard deviation is its square root and is in the original units. Risk managers usually quote standard deviation because it is easier to interpret.

How do I calculate the variance of a discrete random variable?

Compute the mean μ = Σ x·p(x), then E(X²) = Σ x²·p(x). Variance is E(X²) − μ². You can also sum (x − μ)²·p(x), which gives the same answer.

What happens to expected value and variance when I multiply a variable by a constant and add another?

For Y = a + bX, the mean becomes a + b·E(X) and the variance becomes b²·Var(X). The added constant does not change variance. Standard deviation scales by the absolute value of b.

Is the expected value of a sum always the sum of expected values?

Yes. E(X + Y) = E(X) + E(Y) holds whether or not X and Y are independent. The variance of a sum is different, because it also depends on covariance.