Question 1
An insurer covers 10 independent properties. Each of the 10 properties independently has probability 0.10 of a claim this year.
Calculate the probability that at least 2 of the 10 have a claim.
Exam P formulas, and when to use each
No formula sheet is handed out for Exam P, only a normal table. These are the results worth knowing cold, with what each one is for.
| Union | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
|---|---|
| Bayes | P(A | B) = P(B | A)P(A) ÷ P(B) |
| Variance | Var(X) = E[X²] − (E[X])² |
| Deductible | Exponential loss, deductible d: E[payment] = θe^(−d/θ) |
Discrete distributions
| Distribution | P(X = k) | Mean | Variance |
|---|---|---|---|
| Binomial(n, p) | C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ | np | np(1 − p) |
| Poisson(λ) | e^(−λ)λᵏ ÷ k! | λ | λ |
| Geometric(p), failures before the first success | (1 − p)ᵏp | (1 − p) ÷ p | (1 − p) ÷ p² |
| Negative binomial(r, p), failures before the r-th success | C(k + r − 1, k)pʳ(1 − p)ᵏ | r(1 − p) ÷ p | r(1 − p) ÷ p² |
| Hypergeometric(N, K, n) | C(K, k)C(N − K, n − k) ÷ C(N, n) | nK ÷ N | (nK ÷ N)(1 − K ÷ N)(N − n) ÷ (N − 1) |
Continuous distributions
| Distribution | Key fact | Mean | Variance |
|---|---|---|---|
| Uniform(a, b) | F(x) = (x − a) ÷ (b − a) | (a + b) ÷ 2 | (b − a)² ÷ 12 |
| Exponential, mean θ | P(X > x) = e^(−x/θ); memoryless | θ | θ² |
| Gamma(α, θ) | Sum of α exponentials when α is a whole number | αθ | αθ² |
| Normal(μ, σ) | z = (x − μ) ÷ σ, then the table | μ | σ² |
| Beta(a, b) on (0, 1) | Density proportional to xᵃ⁻¹(1 − x)ᵇ⁻¹ | a ÷ (a + b) | ab ÷ ((a + b)²(a + b + 1)) |
Insurance payments
- Deductible d: payment = max(X − d, 0).
- Policy limit u: payment = min(X, u).
- Coinsurance α: the insurer pays the share α of the covered loss.
- Exponential loss: E[min(X, m)] = θ(1 − e^(−m/θ)), so E[max(X − d, 0)] = θe^(−d/θ).
- Given a payment is made, an exponential excess has mean θ again.
Two variables and sums
- Cov(X, Y) = E[XY] − E[X]E[Y], and correlation = Cov ÷ (σX σY).
- Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y).
- Central limit theorem: a total of n independent claims is roughly normal with mean nμ and standard deviation √n·σ.
- Minimum of independent exponentials: exponential, with the rates 1/θ added.
- Maximum of n independent values: P(max ≤ x) = F(x)ⁿ.
Try 3 questions
Question 2
The loss from a theft is exponential with mean $1,000. The policy has a deductible of $250.
Calculate the expected payment by the insurer per loss.
Question 3
Two lines of business have annual losses X and Y. , and the correlation between them is 0.30.
Calculate .
Next step
Questions people ask
Is there a formula sheet on Exam P?
No. The only reference provided is a table of the standard normal distribution, shown on screen. Every other formula has to be known.
Which geometric distribution does Exam P use?
It is defined two ways: the number of failures before the first success, or the number of the trial on which the first success comes. Read which one the question counts, then shift by one if needed.
Are moment generating functions on Exam P?
The suggested readings in the January 2027 syllabus exclude moment generating functions, and the learning outcomes do not list them.