Title: InkMaths / Further Statistics 1: distributions and tests
On every note of this deck, unless a note says otherwise:
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Front: SPOT THE ERROR
Say what is wrong with this claim:
A test at the 5% level is wrong 5% of the time.
Back: The 5% is the chance of rejecting a true null hypothesis, and only when it is true. It says nothing about Type II errors, which are often far more likely, and it is a conditional probability rather than an overall error rate.

Front: STANDARD RESULT — LEARN IT
Geometric:
P(X = x) = ?, μ = ?
Back: p(1 - p)^x-1, μ = 1p, σ^2 = 1-pp^2

Front: STANDARD RESULT — LEARN IT
Variance from a PGF:
Var(X) = ?
Back: G''(1) + G'(1) - [G'(1)]^2

Front: STANDARD RESULT — LEARN IT
Central Limit Theorem:
sample mean ≈ ?
Back: N(μ, σ^2n), standard error σ√n

Front: STANDARD RESULT — LEARN IT
Negative binomial:
rth success on trial x: P = ?
Back: ^x-1C_r-1 p^r(1 - p)^x-r, μ = rp

Front: STANDARD RESULT — LEARN IT
Goodness of fit:
χ^2 = ?
Back: Σ (O_i - E_i)^2E_i

Front: STANDARD RESULT — LEARN IT
Linear transformation:
Var(aX + b) = ?
Back: a^2 Var(X): the shift b contributes nothing

Front: CHECK YOURSELF
A goodness-of-fit test has 6 classes after pooling, with one parameter estimated from the data. State the degrees of freedom.
Back: 6 − 1 − 1 = 4: one lost to the fixed total, one to the estimated parameter.

Front: STANDARD RESULT — LEARN IT
Generating function:
G(t) = ?, mean = ?
Back: E(t^X), mean = G'(1)

Front: STANDARD RESULT — LEARN IT
Poisson:
X is Po(λ): P(X = x) = ?
Back: e^-λλ^xx!, with E(X) = Var(X) = λ

Front: STANDARD RESULT — LEARN IT
Discrete variance:
Var(X) = ?
Back: Σ x^2 P(X = x) - μ^2

Front: DEFINE
Define: power of a test.
Back: The probability of correctly rejecting the null hypothesis at a stated alternative, equal to one minus the probability of a Type II error there.

Front: STANDARD RESULT — LEARN IT
Contingency table:
E_ij = ?, degrees of freedom?
Back: row × columntotal, (r-1)(c-1)