Mathematical Expectation for CA Foundation

Understand Mathematical Expectation

Key formulas to remember

Expected value (discrete)

E(X) = Σ x·p(x) = x₁p₁ + x₂p₂ + … + xₙpₙ

Probabilities must be non-negative and satisfy Σ p = 1.

Expectation of a function

E(X²) = Σ x²·p(x)

Square the value x, not the probability. Do not use [E(X)]² here.

Variance using expectation

Var(X) = E(X²) − [E(X)]²

Standard deviation is the positive square root of the variance.

Expectation of a constant

E(c) = c

A constant always has the same value, so its average is itself.

Linear change

E(aX + b) = a·E(X) + b

Holds for any constants a and b. The constant b is not multiplied by a.

Addition theorem

E(X + Y) = E(X) + E(Y)

Holds for any two random variables, independent or not.

Multiplication theorem

E(XY) = E(X)·E(Y)

Holds only if X and Y are independent.

Variance under linear change

Var(aX + b) = a²·Var(X)

Adding b does not change the spread.

Fair game

Fair if expected gain = 0

Expected gain is the sum of (gain or loss × probability), with losses taken as negative.

How to solve Mathematical Expectation questions

1. 1Identify the random variable X and list every value it can take. Treat losses as negative numbers.
2. 2Write the probability of each value. If a constant such as k is unknown, use Σ p = 1 to find it first.
3. 3Check that the probabilities add to 1 and none is negative.
4. 4Make a table with columns x, p(x) and x·p(x).
5. 5Add the x·p(x) column to get E(X).
6. 6If the question asks for E(X²) or variance, add a column x²·p(x), then use Var(X) = E(X²) − [E(X)]².
7. 7If the question gives a linear form such as aX + b, use E(aX + b) = aE(X) + b instead of building a new table.
8. 8For decisions, compare expected values and choose the better one. For fairness, check whether E equals 0.

Quickest way: Shortcut using properties and option checks

Common mistakes in Mathematical Expectation

• Not checking that probabilities sum to 1 before calculating.

  Students rush to multiply and add, and miss that a constant k is unknown.

  Fix: Write Σ p = 1 as your first line and solve for k before computing E(X).

• Writing E(aX + b) = aE(X) and dropping b.

  Students remember that constants come out, but forget that an added constant stays.

  Fix: Write E(aX + b) = aE(X) + b. Test it with a constant: E(5) must be 5.

• Using E(XY) = E(X)E(Y) without independence.

  It looks like the addition theorem, which always holds.

  Fix: Use the multiplication theorem only when the question says X and Y are independent.

• Computing E(X²) as [E(X)]².

  Squaring after averaging feels the same as averaging squares.

  Fix: Square each value first, multiply by its probability, then add. Expect E(X²) ≥ [E(X)]².

• Ignoring the sign of a loss, or the entry fee.

  Word problems list amounts without signs.

  Fix: Write losses as negative numbers. Subtract any entry fee or cost from each outcome before weighting.

• Treating E(X) as the most likely value.

  The word 'expected' suggests what will probably happen.

  Fix: Remember it is a weighted average, not the mode. It may not be a possible value of X.

Worked examples