Further Statistics - Year 1

Question 1

X = {1, 2, 3, 4, 5, 6}

Why is it considered discrete

Answer 1

It has limited values

Question 2

Give the formula used for the random variable X and xi with probabilities

Answer 2

P(X=xi)

Question 3

Give the term for expectation and mean

Answer 3

Expectation: E(X)
Mean: x bar

Question 4

Give the equation for the Binomial Distribution of X and state what the values mean

Answer 4

X~B(n,p)
X: Discrete random variable
B: The type of equation (Binomial)
n: The number of trials
p: The probability of the event occurring

Question 5

How can the Binomial Distribution be calculated

Answer 5

ⁿCᵣ pˣ(1-p)ⁿ⁻ˣ

Question 6

How can the mean and variance be calculated from a Binomial Distribution

Answer 6

Mean: np
Variance: np(1-p)

Question 7

State the Poisson Distribution for X and state what the values mean

Answer 7

X~Po(λ)
X: Discrete Random Variable
Po: The type of distribution (Poisson)
λ: The mean

Question 8

State the formula for the Poisson Distribution

Answer 8

.
λˣ
e^λ —–
x!

Question 9

State the mean and variance of the Poisson Distribution

Answer 9

Mean: λ
Variance: λ

Question 10

Rearrange E(aX+b)

Answer 10

aE(X)+b

Question 11

State how variance (σ²) is found in terms of expected value

Answer 11

Var(X) = E(X²) - (E(X))²

Question 12

Rearrange Var(aX+b)

Answer 12

a²Var(X)

Question 13

State the equation for x bar

Answer 13

n

Question 14

What is the equation for standard deviation (σ)

Answer 14

.      
      \_\_\_\_\_\_\_\_\_\_\_\_\_
    / Σx²   (  Σx  )²
  /  ------ - ( ------ )
√     n     (   n   )

or

  \_\_\_\_
/ Sₓₓ   /  ------- √      n

Question 15

What is the formula for ⁿCr

Answer 15

.
n!
————
(n-r)! r!

Question 16

X~B(30,p) where p is usually 0.25

Describe the hypothesis test used to find p and find the effective level of significance

Answer 16

H₀: p=0.025
H₁: p>0.025

Test at 5% significance level
P(X≥x)≤0.05
P(X=12)=0.0507, 0.0507>0.05 → Accept H₀
P(X=13)=0.0216, 0.0216<0.05 → Reject H₀

Effective level of significance = 2.16% (P(X=x))

Question 17

What us the expected value in terms of n

Answer 17

2

Question 18

What is the variance in terms of n

Answer 18

2

Question 19

What is the expected value in terms of Σ

Answer 19

.
n
P(X=x) Σ i
i=1

Question 20

What is the variance in terms of Σ

Answer 20

.
n
P(X=x) Σ i²
i=1

Question 21

What would be the result of X~Po(λ) + Y~Po(μ)

Answer 21

(X + Y)~Po(λ +μ)

W~Po(ν)

Question 22

When can X~B(n,p) values be used for X~Po(λ) and how are they related

Answer 22

When n>50 and p<0.1

np = λ