F3 Mathematical Foundations / Probability Theory II

Question 1

What is the expected value?

Answer 1

The mean. Each outcome is weighted by its probability.

Theoretical: E(X)=μ
Empirical: E(X)=x-bar

Question 2

Is the expected value a linear operator?

Answer 2

Yes.

Additivity: E(x+y)=E(x)+E(y)
Homogeneity / scalar multiplication: E(ax)=a*E(x)

Question 3

What is the expected value of a constant?

Answer 3

The constant: E(a)=a

Question 4

What is the difference between the theoretical and empirical expected value?

Answer 4

Theoretical: The sum of all outcomes (x) times their probability P(X=x)

Empirical: The sum of all observations divided by the number of observations.

Question 5

What is the first moment about zero

Answer 5

The mean

Question 6

What is the first central moment?

Answer 6

Always zero as central moments describe the shape of a probability distribution relative the mean of the distribution. Irrelevant

Question 7

What is the second central moment?

Answer 7

Var(x) = [(x - μ_x )^2]

The variance of the distribution, measuring the spread or variability around the mean. We are “punished” exponentially for very big outliers because of the square.

Question 8

What is the lower bound of the variance and what is the variance of a constant?

Answer 8

Var(x) > 0.
Var(a) = 0.

Question 9

What is the third central moment?

Answer 9

Measures the skewness in absolute terms. Normally it’s standardized by the standard deviation:

= 0: symmetrical distribution
>0: Right skewed
<0: Left skewed

Question 10

What is the fourth central moment?

Answer 10

Measures the “tailedness” or “peakedness” of the distribution. Again, we use the standardized kurtosis.

=3: Normal distribution (mesokurtic)
<3: Flatter peak
>3: Sharper peak

Question 11

What is the covariance?

Answer 11

Describes the relation between two variables. means. It measures the joint variability/linear relationship.

The bounds are -∞ to ∞.

Question 12

How is the covariance interpreted?

Answer 12

How variables move around their mean on average.

> 0: When x is above its mean value, we would expect y to be above its mean value as well.

<0: When x is above its mean value, we would expect y to be below its mean value.

=0: Independent

Question 13

What is the equation for the covariance

Answer 13

Cov(X,Y) = E([x - μ_x ] *[y - μ_y ])

Question 14

What is the covariance of Cov(x,x)?

Answer 14

Var(x)

Question 15

What happens if you add a constant to the covariance?

Answer 15

Adding a constant to the variance of something doesn’t change it at all.

Cov(β+x,α+y)=Cov(x,y)

Question 16

What are the bounds of correlation?

Answer 16

-1 to 1. It’s the normalized measure (by standard deviation) of the covariance between to variables.

Question 17

What is a PDF?

Answer 17

Probability density function: The relative frequency or likelihood of drawing observations in and interval (≠ probability).

The PDF allows me to compare the relative likelihood of the random variable falling within different intervals.

If the PDF is higher for one interval compared to another, then the probability of drawing a value from that interval is higher

Question 18

What is the probability of a point estimate in a PDF?

Answer 18

Zero as probability is always defined over an interval in PDF

Question 19

What is the CDF?

Answer 19

What is the probability up to a certain point? The integral of the PDF from upper to lower bound.

The CDF will always be smooth for continuous variables because there is no definitive probabilities for point estimates.

Question 20

Mention five continuous distributions

Answer 20

Uniform
Normal distribution
Standard normal distribution
Logistic
Student t distribution

Question 21

How is the PDF and CDF denoted?

Answer 21

PDF = f(x)
CDF = F(x)

Question 22

What does the CDF for uniform distribution look like?

Answer 22

It’s linear as an increase in x has the same effect anywhere in the distribution.

Question 23

Describe the normal distribution

Answer 23

It’s bell shaped and has two parameters:

μ (location)
σ^2 (scale)

Question 24

Describe the standard normal distribution. Draw the PDF and the CDF.

Answer 24

All normal distributions can be transformed into a standard normal distribution. Denoted ϕ

μ = 0 (location)
σ^2 = 1 (scale)

CDF: Φ(0)=0,5 because the distribution is symmetrical. The highest increase in probability is just around zero.

PDF: Bellshaped
CDF: A hill

Question 25

What is the student t distribution used for?

Answer 25

Used for testing the statistical significance of our beta coefficient. Critical value ±1,96.

Question 26

How do we obtain a large t-value?

Answer 26

Either by a large coefficient or by a small standard error.

What makes the standard error smaller?
(1) Big sample size
(2) Low variance in regression line (SSR)
(3) High variance in x-values