Further Statistics 1

Formula for E(X)

E(X) = Σ x·P(X=x)

Formula for Var(X)

Var(X) = E(X²) − [E(X)]² where E(X²) = Σ x²·P(X=x)

E(aX + b) ? E(X + Y) ?

E(aX + b) = aE(X) + b

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

Var(aX + b)?

Var(aX + b) = a²·Var(X)

E(g(X)) =?

E(g(X)) = Σ g(x)·P(X=x)

Steps to find E(X) or Var(X)

1. Use E(X)=Σxp or Var=E(X²)−[E(X)]²

2. Or build new probability distribution then apply formulas

Steps to find X given Y=g(X)

1. Use rules E(aX+b), Var(aX+b)

2. Rearrange formula or set up prob distribution

Poisson PMF

P(X=x) = e^{-λ}·λˣ / x! , x=0,1,2,…

Conditions for Poisson model

Events occur independently, singly, at a constant average rate (mean ∝ interval length)

Mean and Variance of X~Po(λ)

E(X) = λ

Var(X) = λ | Mean = Variance

Sum of independent Poissons

X~Po(λ) and Y~Po(μ), independent → X+Y ~ Po(λ+μ)

Must be same interval

Poisson table probability forms

P(X≤k): read directly | P(X<k) = P(X≤k−1) | P(X≥k) = 1−P(X≤k−1) | P(X>k) = 1−P(X≤k)

Binomial → Poisson approximation conditions

X~B(n,p), n large, p small → X ≈ Po(np)

Generally np≤10

Geometric PMF and 4 conditions

P(X=x) = p(1−p)^{x−1}, x=1,2,3,…

Conditions: independent trials, constant p, counting trials to 1st success

Geometric cumulative formulas

P(X≤x) = 1−(1−p)^x

P(X>x) = (1−p)^x

P(X<x) = (1−p)^{x−1}

Mean & Variance: X~Geo(p)

E(X) = 1/p

Var(X) = (1−p)/p²

Negative Binomial PMF

P(X=x) = C(x−1, r−1)·p^r·(1−p)^{x−r} | x = r, r+1, r+2,…

Last trial must be the success

Mean & Variance: X~NB(r,p)

E(X) = r/p

Var(X) = r(1−p)/p² | (Geometric × r)

Steps: Negative Binomial problems

1. Identify r and p

2. Use PMF

3. For P(X≤x) or P(X≥x): switch to Binomial

4. If given attempt k was success: use NB(r−1,p) for remaining

5. Last trial must always be the success

Steps: Poisson hypothesis test (p-value)

1. Define X

2. State H₀: λ=λ₀, H₁

3. Assume H₀ true → X~Po(λ₀)

4. Find observed x

5. Calculate probability

6. Compare to α

7. Reject H₀ if p<α; conclude in context

Steps: finding critical region (Poisson)

Decrease → P(X≤c) < α, largest c

Increase → P(X≥c) < α, smallest c

P(X≥c) = 1−P(X≤c−1)

Two-tailed test rule + actual significance

Split α: use α/2 for each tail

Actual significance = P(lower region) + P(upper region)

Steps: Geometric hypothesis test

1. Define X~Geo(p), state H₀, H₁

2. Geo critical regions are opposite: H₁:p>p₀ → lower tail

3. Use Geo cumulative formulas

4. Compare to α; conclude

Critical region decision rule

X in critical region → Reject H₀

X outside → Do not reject H₀

Actual significance = P(in critical region | H₀ true)

CLT statement

X₁,…,Xₙ random sample, mean μ, variance σ² → X̄ ≈ N(μ, σ²/n) for large n

Mean & Variance by distribution

B(n,p): μ=np, σ²=np(1−p)

Geo(p): μ=1/p, σ²=(1−p)/p²

NB(r,p): μ=r/p, σ²=r(1−p)/p²

Po(λ): μ=λ, σ²=λ

Steps: CLT probability problem

1. Identify distribution; find μ and σ²

2. Identify n

3. X̄~N(μ, σ²/n)

4. Convert question to statement about X̄

5. Use normal distribution on calc | No continuity correction needed

Chi-squared test statistic

X² = Σ(O−E)²/E or Σ O²/E − N

O=observed, E=expected, N=total

Degrees of freedom: goodness of fit

ν = cells (after combining) − constraints

Uniform: ν=cells−1

Binomial/Poisson (param estimated): ν=cells−2

Param not estimated: ν=cells−1

Rule for combining cells

If any expected frequency < 5, combine adjacent cells until all E≥5

Decision rule: goodness of fit

X²_calc < X²_crit → Accept H₀ (good fit)

X²_calc > X²_crit → Reject H₀ | Always one-tailed

Steps: goodness of fit test

1. State H₀/H₁

2. Set significance level

3. Estimate parameters if needed

4. Calculate expected frequencies

5. Combine cells (E<5)

6. Find ν and critical value

7. Calculate X²

8. Compare and conclude

Contingency table: expected frequency & df

E = (row total × column total) / grand total

ν = (rows−1)(columns−1)

H₀: variables are independent

Definition of PGF

Gₓ(t) = E(tˣ) = Σ P(X=x)·tˣ

Coefficient of tˣ = P(X=x)

Gₓ(1) = 1 always

PGFs for standard distributions

B(n,p): (1−p+pt)ⁿ

Po(λ): e^{λ(t−1)}

Geo(p): pt/(1−(1−p)t)

NB(r,p): [pt/(1−(1−p)t)]^r

Mean & Variance from PGF

E(X) = G'ₓ(1)

Var(X) = G''ₓ(1) + G'ₓ(1) − [G'ₓ(1)]²

P(X=n) = (1/n!)·G^(n)ₓ(0)

PGF of sum of independent variables

Z = X+Y (independent) → Gz(t) = Gₓ(t) × G_Y(t)

PGF of linear transformation Y=aX+b

G_Y(t) = tᵇ · Gₓ(tᵃ)

Steps: derive PGF from first principles

1. Write P(X=x)

2. Use Gₓ(t) = Σ P(X=x)·tˣ

3. Sub probability formula

4. Recognise series: geometric → Geo, binomial expansion → Binomial, eˣ series → Poisson

5. Simplify

Type I and Type II errors

Type I: Reject H₀ when H₀ TRUE

P(Type I) = actual significance level

Type II: Accept H₀ when H₀ FALSE

P(Type II) found using true parameter value

Power of a test

Power = 1 − P(Type II error) = P(reject H₀ | H₀ false)

Power > 0.5 → more likely correct conclusion

Size of a test

Size = P(Type I error) = P(X in critical region | H₀ true) = actual significance level

Steps: find P(Type II error)

1. Find critical region

2. Invert to get acceptance region

3. Replace H₀ param with TRUE value

4. P(Type II) = P(X in acceptance region | true param)

Power function steps

1. Write Power = P(X in critical region)

2. Replace H₀ param with unknown θ

3. Express as formula in θ — do NOT sub a number

4. Verify: subbing H₀ value should equal size

5. Compare tests: recommend higher power