Statistical Inference

Question 1

Sample mean

Answer 1

x¯ = 1/n ∑(i=1, n) xᵢ

Question 2

Sample variance

Answer 2

s² = 1/(n-1) ∑(i=1 ,n) (xᵢ - x¯)²

Question 3

Expected value

Answer 3

The expected value of a random variable X is a measure of the centre or average value that one would expect to occur if the random experiment or process were repeated many times. E(X)= ∑ xi ⋅P(X=xi)

Question 4

Expected value of the sample mean

Answer 4

E[X¯] = µ

Question 5

Sampling distribution

Answer 5

A probability distribution for a sample statistic

Question 6

Variance

Answer 6

Var[x] = E[(x - µ)²] or Var[x] = E[x²] - E[x]²

Question 7

Probability density function

Answer 7

Probability density function assigns a probability to intervals of values compared to the probability mass function which assigns probabilities to discrete values
PDF is usually denoted f(x)

Question 8

Cumulative density function

Answer 8

CDF is denoted F(x) and is the antiderivative of the PDF f(x)
F(x) = P(X ≤ x)

Question 9

PDF of sample maximum

Answer 9

for, Z = max(X₁ + X₂ + … + Xₙ)
g(z) = nf(z) (F(z))^n-1

Question 10

PDF of sample minimum

Answer 10

for, W = min(X₁ + X₂ + … + Xₙ)
h(w) = nf(w) (1 - F(w))^n-1

Question 11

CDF of sample maximum

Answer 11

G(z) = (F(z))ⁿ or the integral of g(z)

Question 12

CDF of sample minimum

Answer 12

H(w) = 1 - (1 - F(W)ⁿ or the integral of h(w)

Question 13

Normal distribution

Answer 13

Let X be a normally distributed random variable with mean µ and variance 𝜎².
then we have X ∼ N[ µ, 𝜎²]

Question 14

Standardized normal random variable

Answer 14

Let X be a normally distributed random variable with mean µ and variance 𝜎².
Then Z = (X - µ)/ 𝜎
and we have mean = 0 and variance = 1

Question 15

What is zᵧ, Used to determine the value of x in P(X < x)

Answer 15

we denote zᵧ to be the scalar such that P[ Z < zᵧ] = γ. where Z is a random variable that has a standard normal distribution, Z ∼ N [0, 1].

Question 16

when should you use the student’s t-distribution?

Answer 16

• When the sample is small ie. n ≤ 30
  or when the sample standard deviation is given instead of the population standard deviation.

Question 17

Student’s t-distribution

Answer 17

If X₁, X₂, …, Xₙ is a random sample from a normal distribution with mean µ and variance 𝜎² then the random variable T = (X¯ - µ)/(S/√n) has the t-distribution with (n-1) degrees of freedom
T∼ t (n-1)

Question 18

what is tᵧ(n-1),

Answer 18

We denote tᵧ (n-1) to be the scalar such that P[ T < tᵧ(n-1)] = γ where T ∼ t(n-1)
For example,
n = 11, P[ T < 1.812] = 0.95 t0.95((11)-1) = 1.812

Question 19

Chi-squared distribution

Answer 19

The chi-squared distribution is the distribution of the random variable Z²₁ + Z²₂ + . . . + Z²ₙ
If Z is a random variable with chi-squared distribution we can write Z ∼ X²(ν) where ν is the degree of freedom
and X² = (n-1)S²/(𝜎²)
µ = v and 𝜎² = 2v

Question 20

How can we tell its a chi-squared distribution?

Answer 20

When the random variable X follows the sum of the squares of independent standard normal random variables.
X = Z²₁ + Z²₂ + . . . + Z²ₙ

Question 21

How can we tell its an F-distribution?

Answer 21

The f-distribution is characterised by 2 different sets of degrees of freedom v₁ and v₂.
Use f-distribution when we have two independent random variables X and Y such that X∼ X²(ν₁) and Y ∼ X²(ν₂)
we have F = (X/v₁)/(Y/v₂) of f-distribution F ∼ F(ν₁, ν₂)

Question 22

What is the z score of a poppulation

Answer 22

Z = (X -X¯)/(√ (𝜎²/n))

Question 23

Central limit theorem

Answer 23

Let X be an (iid) random variable with mean µ and variance 𝜎²
then we have
(a) ∑(i=1, n) Xᵢ ∼ N [nµ , n𝜎²], approximately
(b) X¯ ∼ N[µ , 𝜎²/n], approximately

Question 24

what is an estimator

Answer 24

an estimator denoted θ̂ is a statistic used to estimate θ.

Question 25

What is an unbiased estimator

Answer 25

an estimator θ̂ of θ is said to be unbiased if E[ θ̂ ] = θ. otherwise it is biased with bias[ θ̂ ] = E[ θ̂ ] - θ.

Question 26

What is the mean square error of an estimator.

Answer 26

The mean square error (MSE) of an estimator θ̂ of θ is MSE[θ̂] = E[ (θ̂ - θ)²]
or
MSE[θ̂ ] = Var[ θ̂ ] + {bias[ θ̂ ]}²

Question 27

Better estimator

Answer 27

Let θ̂₁ and θ̂₂ be two estimators of a parameter θ,
θ̂₁ is said to be a better estimator in MSE than θ̂₂ if MSE[ θ̂₁ ] < MSE [ θ̂₂ ].

Question 28

Determine the z score of a confidence interval

Answer 28

For a 100(1 - 𝛼 ) % confidence level we take z(1-(𝛼/2))
ie. for a 95% confidence level 𝛼 = 0.05.
z(1-(0.05/2)) = Z0.975
then from the z distribution tables find the area of the normal distribution graph to the left of our z𝛼 value.
so z(1-(0.05/2)) = 1.645.

Question 29

Confidence intervals for means, with normally distributed population and known variance

Answer 29

P[¯X - Z(1-(𝛼/2)) (𝜎/√ n) < µ < ¯X + Z(1-(𝛼/2)) (𝜎/√ n)] = 1 - 𝛼
where 𝛼 is the amount of risk
or
C = ¯X ± Z(1-(𝛼/2)) (𝜎/√ n)

Question 30

Confidence intervals for means, with normally distributed population and unknown variance

Answer 30

P[¯X - t((𝛼/2),n-1) (s/√ n) < µ < ¯X + t((𝛼/2),n-1) (s/√ n)] = 1 - 𝛼
(where s = sample standard deviation.
or
C = ¯X ± t((𝛼/2),n-1) (s/√ n)

Question 31

Poisson distribution

Answer 31

Probability mass function for possion distribution is given by P( X = x ) = (𝜆^(x) e^(-𝜆))/x!
The mean and variance for the poisson distribution is 𝜆.
E[ X ] = 𝜆 Var[ X ] = 𝜆

Question 32

Confidence intervals for variance

Answer 32

P[S² - S²/(X²(1-(𝛼/2)) < 𝜎² < S² + S²/(X²(𝛼/2)] = 1 - 𝛼