lecture 2

Question 1

If X is a continuous random variable, What is P(X=1)?

Answer 1

An accurate (but unhelpful) answer to this question
is P(X=1)=0. In fact 𝑃(𝑋=𝑎)=0 at any given a within the interval.
- Why? - as it is just one point we cannot calculate the area - it is an infinitesimally small point with such as infinitesimally small area which we say is 0

∫_-∞^∞f(x)=1 (total probability)

Question 2

What is a continuous random variable?

Answer 2

is a random variable where the data can take infinitely many values.

for example if we want to measure the temprature of a room, the amount of possible value is infinite

Question 3

What criteria must you satisfy to become a probability density function?

Answer 3

PDF example –> p(a ≤ X ≤ b) = ∫_a^b f(x) dx

(1) f(x) must be nonnegative for each value of the random variable, and
- f(x) ≥ 0

(2) the integral over all values of the random variable must equal one.
- ∫_-∞^∞f(x)dx = 1

Question 4

What is the z transformation or standardising a normal distribution?

Answer 4

• It is possible to transform any normal distribution into the standard normal distribution (mean = 0, standard deviation = 1) as follows:
• X~N(μ,σ^2)
• Z=(X-μ)/σ ~ N(0,1)
• This is useful because we can take data from different sources onto the same scale and only have to tabulate the standard normal distribution to be able to look up critical values and/or p-values for test statistics.

Question 5

What is the function normal distribution?

Answer 5

• f(x) = (1/σsqrt(2π)) * exp[-((x-μ)^2)/(2σ^2))]

Question 6

What are some examples of higher order moments?

Answer 6

Higher order moments are often scaled by the standard deviation to obtain measures such as:

• skewness –> E(X - E(X))^3/σ^3
• kurotosis –> E(X - E(X))^4/σ^4
• These measures are useful in characterising the shape of a distribution and are often referred to as the moments of the distribution even though, strictly speaking, they are transformations of the raw moments.

Question 7

What is the mean function for continuous distribution

Answer 7

intergral f(x)*X

Question 8

What is the variance function for continuous distribution?

Answer 8

integral (x–μ)^2 *f(x)

Question 9

How do you find out the function of higher order moments for continuous distribution?

Answer 9

• can be calulcuated by integrating a function of the form

- ∫_a^bf(x) (x-E(x))^k *f(x) dx (where a and b are the minimum and maximum possible values) and scaling by σ^k

Question 10

What is the Chi-squared distribution?

Answer 10

Is used to test the ‘goodness of fit’ a theoretical model is to a observed one

• so looking at the variance of the risidual error (risdual sum of squares) or a regression model for the actual data
• Also looking at the probability you could get those errors while holding some variables constant.
• The Chi-squared is derived or sampled from normal distribution with the formula:
• χ = Σ_j=1^k (Z_j^2)
• Z is the risdual of a model
• the random variable defined by this is said to follow a chi-squared distribution with k degrees of freedom.
• When looking up on a table the P-value is the probability of it be larger than that value

Question 11

What does the Chi-squared distribution look like on a graph?

Answer 11

if degrees of freedom (k) = 01 or 2 –> chi squared no longer has a PDF which takes the value 0 at x=0 –> instead the value of the PDG tends to infinity as x(chi) tends to zero –> look like 1/x graph for k=1 and is downwards sloping for k=2
- if k > 2 –> The PDF takes the value 0 for x=0, reaches a single peak for some value of x >0 and declines asymptotically to 0 as x becomes large. - it is positively skewed.

Question 12

What is the mean and variance for Chi-squared distribution?

Answer 12

mean –> k

variance –> 2k

Question 13

What happens when k is large for a Chi-squared distribution?

Answer 13

k becomes large, the asymmetry which we have
observed in the PDF of the chi-squared distribution becomes less pronounced.
becomes more normal and symmetrical as more observations are recorded

Question 14

What is the F-distribution?

Answer 14

Suppose we have two random variables each of which follows a chi-squared distribution
- A variable follows an F-distribution if it is constructed as the ratio of two Chi-squared distributed variables each of which is divided by its degrees of freedom:

the value given in the table give you F values which if are greater than the critical value (usually 5%) it is rejected

Question 15

What is the Student’s t-distribition?

Answer 15

• Student’s t distribution is often referred to simply as the t distribution. It arises in
  econometrics (and in many other statistical situations) when we wish to conduct hypothesis tests on a variable which we assume is normally distributed but for which we do not know `the variance