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 level the test is set to. The chance of rejecting a true null is the test's size, the probability of the critical region given H₀, and a discrete test lands at or below the nominal 5%. 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:
X is Geo(p), x = 1, 2, 3, …: P(X = x) = ?, μ = ?, σ^2 = ?\(X \text{ is Geo}(p), \, x = 1, 2, 3, …: \text{P}(X = x) = ?, μ = ?, σ^{2} = ?\)X is Geo(p), x = 1, 2, 3, …: P(X = x) = ?, μ = ?, σ^2 = ?
Back: p(1 - p)^x-1, 1p, 1-pp^2\(p(1 - p)^{x-1}, \frac{1}{p}, \frac{1-p}{p^{2}}\)p(1 - p)^(x - 1), (1/p), ((1 - p)/p^2)

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

Front: RECALL
State what the Central Limit Theorem buys you.
Back: For a large enough sample of independent observations from one distribution of finite variance, the mean is approximately normal whatever that population looks like, with mean μ and variance σ²/n. The mean and variance hold for any n. Normal is the theorem's whole contribution, so quote the sample size when you lean on it.

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

Front: STANDARD RESULT — LEARN IT
Goodness of fit:
χ^2 = ?\(χ^{2} = ?\)χ^2 = ?
Back: Σ (O_i - E_i)^2E_i\(Σ \frac{(O_{i} - E_{i})^{2}}{E_{i}}\)Σ (((O_i - E_i)^2)/E_i)

Front: RECALL
State how a Poisson hypothesis test differs from a binomial one.
Back: The machinery is identical, only the distribution under the null changes. Test λ rather than p, and scale λ to the interval the question actually observes before you touch the tables. For a geometric model the tail is easy, since P(X ≥ x) is just (1 − p) to the power x − 1.

Front: STANDARD RESULT — LEARN IT
Linear transformation:
Var(aX + b) = ?\(\text{Var}(aX + b) = ?\)Var(aX + b) = ?
Back: a^2 Var(X). The shift b contributes nothing\(a^{2} \text{Var}(X). \text{ The shift b contributes nothing}\)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 degree of freedom goes to the fixed total and one to the estimated parameter.

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

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

Front: STANDARD RESULT — LEARN IT
Discrete variance:
Var(X) = ?\(\text{Var}(X) = ?\)Var(X) = ?
Back: Σ x^2 P(X = x) - μ^2\(Σ x^{2} \text{P}(X = x) - μ^{2}\)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?\(E_{ij} = ?, \text{degrees of freedom?}\)E_ij = ?, degrees of freedom?
Back: row × columntotal, (r-1)(c-1)\(\frac{\text{row} × \text{column}}{\text{total}}, (r-1)(c-1)\)((row × column)/total), (r - 1)(c - 1)