Further Statistics

Question 1

1. Discrete: Expectation

Answer 1

E(X) = mean = μ = ∑xP(X=x)

Question 2

1. Discrete: Variance

Answer 2

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

Question 3

1. Discrete: Coding mean - Y = g(X); Y = aX + b

Answer 3

E(g(X)) = ∑g(X)P(X=x)
E(aX+b) = aE(X) + b

Question 4

1. Discrete: Example - Y=X/50 -3. Find E(X) given E(Y)=5.1

Answer 4

Y = X/50 -3
X = 50Y + 150
E(X) = E(50Y + 150) = 50E(Y) + 150 = 50*5.1 +150 = 405

Question 5

1. Discrete: Coding variance - Y = aX + b

Answer 5

Var(Y) = a^2Var(X)

Question 6

1. Discrete: Example - Y=X/50 -3. Find Var(X) given Var(Y)=2.5

Answer 6

Var(Y) = (1/50)^2Var(X)
Var(X) = 2500*Var(Y) = 6250

Question 7

1. Discrete: E(X + Y) =

Answer 7

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

Question 8

2. Poisson: Distribution parameter and probability function

Answer 8

X ∼ Po (λ) per unit time/space, where λ is the rate

P(X=x) = ((e^-λ)*λ^x)/x!

Question 9

2. Poisson: Conditions for events

Answer 9

Independent
Rare/occur singly in time or space
Constant rate - mean number proportional to given interval

Question 10

2. Poisson: Combining two independent variables (Z = X + Y, where X∼Po(λ) and Y∼Po(μ))

Answer 10

X + Y ∼ Po(λ + μ)

Question 11

2. Poisson: Mean and variance

Answer 11

E(X) = Var(X) = λ

Question 12

2. Poisson: Binomial approximation (mean)

Answer 12

X ∼ B (n, p)
X ∼ Po (λ) where λ = np
Because E(X) = np (binomial) = λ (poisson)

Question 13

2. Poisson: Binomial approximation (conditions)

Answer 13

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

Question 14

3. Geo/NB: Geometric distribution and parameter

Answer 14

X ∼ Geo (p), where p is the probability of success and X is the number of trials needed to get one success.

Question 15

3. Geo/NB: Geometric probability function

Answer 15

P(X=x) = p(1-p)^(x-1)

Question 16

3. Geo/NB: Conditions for geometric

Answer 16

Constant probability of success

Independent trials

Question 17

3. Geo/NB: Geometric sequence for Geo distribution (cumulative distribution)

Answer 17

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

Question 18

3. Geo/NB: Cumulative Geo facts (4)

Answer 18

P(X<=x) = 1 - (1-p)^x
P(X>x) = (1-p)^x
P(X>=x) = (1-p)^(x-1)
P(X

Question 19

3. Geo/NB: Mean and variance of geometric

Answer 19

E(X) = 1/p
Var(X) = (1-p)/p^2

Question 20

3. Geo/NB: Negative binomial distribution and parameters

Answer 20

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

Question 21

3. Geo/NB: NB probability function

Answer 21

(x-1)C(r-1) * (p^r)((1-p)^x-r)

Question 22

3. Geo/NB: NB probability function in words

Answer 22

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

Question 23

3. Geo/NB: Mean and variance of negative binomial

Answer 23

E(X) = r/p
Var(X) = r(1-p)/p^2