Example 1: Why 1.96 gives a 95% interval
Using A(1.96) = 0.4750, find P(−1.96 < Z < 1.96).
- The curve is symmetric about 0
P(−1.96 < Z < 1.96) = 2A(1.96)= 2 × 0.4750 = 0.95
Answer: 0.95
DSE M1 Formula Sheet: Calculus and Statistics
This page lists the formulas used in DSE M1, grouped by area. In the examination a standard normal distribution table is provided; plan to know the other formulas by heart. The table gives A(z) = P(0 ≤ Z ≤ z).
Calculus Formulas (DSE M1)
Binomial expansion
(a + b)ⁿ = Σ C(n, r) aⁿ⁻ʳ bʳExponential series
eˣ = 1 + x + x²/2! + x³/3! + …Derivatives
(xⁿ)′ = nxⁿ⁻¹, (eˣ)′ = eˣ, (ln x)′ = 1/x, (aˣ)′ = aˣ ln aProduct / quotient
(uv)′ = u′v + uv′, (u/v)′ = (u′v − uv′)/v²Chain rule
dy/dx = (dy/du)(du/dx)Integrals
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, ∫ (1/x) dx = ln|x| + C, ∫ ekx dx = ekx/k + CDefinite integral
∫ab f(x) dx = F(b) − F(a)Trapezoidal rule
(h/2)[y₀ + 2(y₁ + … + yn−1) + yn], h = (b − a)/nProbability and Discrete Distributions
- Conditional probability:
P(A | B) = P(A ∩ B) / P(B) - Bayes' theorem:
P(B | D) = P(D | B)P(B) / [P(D | A)P(A) + P(D | B)P(B)] - Expectation:
E(X) = Σ x P(X = x) - Variance:
Var(X) = E(X²) − [E(X)]² - Linear change:
E(aX + b) = aE(X) + b,Var(aX + b) = a²Var(X) - Binomial
B(n, p):P(X = k) = C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ, meannp, variancenp(1 − p) - Poisson
Po(λ):P(X = k) = e−λλᵏ / k!, mean λ, variance λ
Normal Distribution, Sampling and Confidence Intervals
- Standardisation:
Z = (X − μ)/σ, whereX ~ N(μ, σ²) - Table:
A(z) = P(0 ≤ Z ≤ z);P(Z > z) = 0.5 − A(z)forz > 0 - Sample mean:
X̄ ~ N(μ, σ²/n) - 95% confidence interval for μ:
x̄ ± 1.96 σ/√n - 99% confidence interval for μ:
x̄ ± 2.575 σ/√n
The values 1.96 and 2.575 come from the table: A(1.96) = 0.4750 and A(2.575) = 0.4950.
Worked Examples — M1 Formula Sheet
Frequently Asked Questions
A standard normal distribution table is provided. Do not count on the calculus and probability formulas on this page being printed; learn them.
1.96, because A(1.96) = 0.4750 and twice that is 0.95. For 99% the value is 2.575.
Binomial B(n, p): mean np, variance np(1 − p). Poisson Po(λ): mean λ, variance λ.