F3 Mathematical Foundations / Probability Theory II Flashcards
What is the expected value?
The mean. Each outcome is weighted by its probability.
Theoretical: E(X)=μ
Empirical: E(X)=x-bar
Is the expected value a linear operator?
Yes.
Additivity: E(x+y)=E(x)+E(y)
Homogeneity / scalar multiplication: E(ax)=a*E(x)
What is the expected value of a constant?
The constant: E(a)=a
What is the difference between the theoretical and empirical expected value?
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.
What is the first moment about zero
The mean
What is the first central moment?
Always zero as central moments describe the shape of a probability distribution relative the mean of the distribution. Irrelevant
What is the second central moment?
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.
What is the lower bound of the variance and what is the variance of a constant?
Var(x) > 0.
Var(a) = 0.
What is the third central moment?
Measures the skewness in absolute terms. Normally it’s standardized by the standard deviation:
= 0: symmetrical distribution
>0: Right skewed
<0: Left skewed
What is the fourth central moment?
Measures the “tailedness” or “peakedness” of the distribution. Again, we use the standardized kurtosis.
=3: Normal distribution (mesokurtic)
<3: Flatter peak
>3: Sharper peak
What is the covariance?
Describes the relation between two variables. means. It measures the joint variability/linear relationship.
The bounds are -∞ to ∞.
How is the covariance interpreted?
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
What is the equation for the covariance
Cov(X,Y) = E([x - μ_x ] *[y - μ_y ])
What is the covariance of Cov(x,x)?
Var(x)
What happens if you add a constant to the covariance?
Adding a constant to the variance of something doesn’t change it at all.
Cov(β+x,α+y)=Cov(x,y)
What are the bounds of correlation?
-1 to 1. It’s the normalized measure (by standard deviation) of the covariance between to variables.
What is a PDF?
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
What is the probability of a point estimate in a PDF?
Zero as probability is always defined over an interval in PDF
What is the CDF?
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.
Mention five continuous distributions
Uniform
Normal distribution
Standard normal distribution
Logistic
Student t distribution
How is the PDF and CDF denoted?
PDF = f(x)
CDF = F(x)
What does the CDF for uniform distribution look like?
It’s linear as an increase in x has the same effect anywhere in the distribution.
Describe the normal distribution
It’s bell shaped and has two parameters:
μ (location)
σ^2 (scale)
Describe the standard normal distribution. Draw the PDF and the CDF.
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
What is the student t distribution used for?
Used for testing the statistical significance of our beta coefficient. Critical value ±1,96.
How do we obtain a large t-value?
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
