Further Statistics Flashcards
- Discrete: Expectation
E(X) = mean = μ = ∑xP(X=x)
- Discrete: Variance
Var(X) = σ^2 = square of standard deviation Var(X) = E(X-E(X))^2) = the mean of the differences between the values and the mean all squared Var(X) = E(X^2) - (E(X))^2 = the mean of the squares minus the square of the means
- Discrete: Coding mean - Y = g(X); Y = aX + b
E(g(X)) = ∑g(X)P(X=x) E(aX+b) = aE(X) + b
- Discrete: Example - Y=X/50 -3. Find E(X) given E(Y)=5.1
Y = X/50 -3 X = 50Y + 150 E(X) = E(50Y + 150) = 50E(Y) + 150 = 50*5.1 +150 = 405
- Discrete: Coding variance - Y = aX + b
Var(Y) = a^2Var(X)
- Discrete: Example - Y=X/50 -3. Find Var(X) given Var(Y)=2.5
Var(Y) = (1/50)^2Var(X) Var(X) = 2500*Var(Y) = 6250
- Discrete: E(X + Y) =
E(X+Y) = E(X) + E(Y)
- Poisson: Distribution parameter and probability function
X ∼ Po (λ) per unit time/space, where λ is the rate
P(X=x) = ((e^-λ)*λ^x)/x!
- Poisson: Conditions for events
Independent
Rare/occur singly in time or space
Constant rate - mean number proportional to given interval
- Poisson: Combining two independent variables (Z = X + Y, where X∼Po(λ) and Y∼Po(μ))
X + Y ∼ Po(λ + μ)
- Poisson: Mean and variance
E(X) = Var(X) = λ
- Poisson: Binomial approximation (mean)
X ∼ B (n, p)
X ∼ Po (λ) where λ = np
Because E(X) = np (binomial) = λ (poisson)
- Poisson: Binomial approximation (conditions)
Var(X) = np(1-p) (binomial) = λ (poisson)
λ is defined as np (so 1-p ≈ 1 ∴ p must be small)
Conditions = large n, small p
- Geo/NB: Geometric distribution and parameter
X ∼ Geo (p), where p is the probability of success and X is the number of trials needed to get one success.
- Geo/NB: Geometric probability function
P(X=x) = p(1-p)^(x-1)
- Geo/NB: Conditions for geometric
Constant probability of success
Independent trials
- Geo/NB: Geometric sequence for Geo distribution (cumulative distribution)
a = 1st term = p r = common ratio = 1-p = q Sn = sum to n terms = a(1-r^n)/(1-r) so P(X<=x) = 1 - (1-p)^x
- Geo/NB: Cumulative Geo facts (4)
P(X<=x) = 1 - (1-p)^x
P(X>x) = (1-p)^x
P(X>=x) = (1-p)^(x-1)
P(X
- Geo/NB: Mean and variance of geometric
E(X) = 1/p Var(X) = (1-p)/p^2
- Geo/NB: Negative binomial distribution and parameters
X ∼ NB (r, p), where r is the number of successes, p is the fixed probability of success and X is the number of trials
- Geo/NB: NB probability function
(x-1)C(r-1) * (p^r)((1-p)^x-r)
- Geo/NB: NB probability function in words
The probability of r-1 successes in x-1 trials (binomial section where n= x-1, p=p and value entered = r-1) * probability of success in xth trial
- Geo/NB: Mean and variance of negative binomial
E(X) = r/p Var(X) = r(1-p)/p^2
