F2 Mathematical Foundations / Probability Theory I Flashcards
What is the formula for the variance of x
Var(x)=E[(x-my_x )^2]
What is the formula for the mean of x
E(x)=(∑x_i )/n
What are three types of probability theory?
1) Classical / frequentist
2) Empirical / bayesian
3) Subjective
What is the difference between probabilistic and deterministic theories?
Probabilistic: If A then B is likely (used in political science)
Deterministic: If A then B.
What is a distribution?
A set of values that a random variable may take. Each values has a weight (=probability)
Which of the following a linear operator?
- Variance
- Expected value
- Integral
- Variance (no)
- Expected value (yes)
- Integral (yes)
What is a constraint for the CDF?
It is non-decreasing. If x increases then the probability will always increase or be the same.
What is a constraint of the PDF?
It is no defined for negativ values
f(x)>0
What is an interger?
Integers: {..-2,-1,0,1,2…}
What is the difference between a scalar, a vector and a matrix?
Scalar: One value
Vector: One dimensional/combination of values
Matrix: Two dimensional (can also be a vector)
What is the difference between an explicit and implicit function?
Explicit: y = x
Implicit: y = f(x)
How do you interpret the beta-koefficent?
With Δx what is the average increase in y?
What is euler’s number? What is it raised to the power of 1 and 0?
2.7182
e^1 = 2.7182
e^0 = 1
What does factorial mean? And what is 0! and 1!
The product of all positive integers from 1 to x (with x!). 5!=54321=120
0! = 1
1! = 1
How does the sum symbol work? Draw it
Index of summation: i (integer values)
Lower limit: m
Upper limit: n
The expression to be summed / element: f(i)
How it works:
Initialization: Start with i=m
Iteration: Increment i with 1 after each step
Termination: Stop once i exceeds n
Summation: Add the values of f(i) for each i from m to n
What are four relevant words of probability theory?
Element of: ∈
Sample space: Entirety of all outcomes (S)
Outcome: Smallest subunit
Event: Partitions of the sample space (S)
What is and intersection and union? Draw a Venn diagram
Intersection (A and B): P(A∩B)
Union (A or B): P(A∪B)
How do you write discrete values? How do you indicate whether endpoints are included or not?
Endpoints not included: (0,1)
Endpoints included: [0,1]
Discrete values: {0,1}
What is the complement of A?
Everything that is not A
How can you calculate the intersection of A and B?
P(A∩B) = P(B|A)P(A) = P(A|B)P(B)
How do you calculate the union of A and B?
P(A∪B) = P(A)*P(B) - P(A∩B)
If A and B are independent:
P(A∩B) = P(A)*P(B)
Define independence and draw it in a regression
The occurrence of A does not depend on B. Probability of A is independent from B.
P(A|B)=P(A)
Regression line is flat.
When are to sets mutually exclusive?
Two events are mutually exclusive when one cannot occur if the other has occurred.
A form of total dependence (no elements in their intersection).
P(A|B)=0
P(A∪B) = P(A) + P(B)
What does collective exhaustivity means?
Each and every event fits into at least one of the categories. Not the same as independence!
P(A)+P(B) = 1
What is DGP?
Data generating proces
What are important laws of probability?
Non-negativity: Probability must be positive
Normalization: The sample space =1
What does discrete variabel mean?
It’s countable.
What is countable infinity?
There is no upper limit but it’s still counted
Define a random variabel.
A variable that can take on array of values with the probability that it can take on any value defined according to a random process (a process we can model)
What do we assume about the error term?
Epsilon is normally distributed ε~N(0,σ^2).
What is a PMF? What is the input and output?
A probability mass function.
Input: Discrete value
Output: Probability for the value [0,1]
Following the law of normalization all possible outcomes sum to 1
What is a CDF for discrete variables? What is the input and output?
A cumulative distribution function - probability summed up to a certain value
Input: Discrete value
Output: Probability of the cumulative probability up until the value [0,1]
Probability will always increase in vertical steps (the staircase).
What is a parameter?
A parameter describe the shape of a distribution e.g. the location parameter (mean/my) and the scale/dispersion (variance/sigma).
Mention four discrete distributions
Uniform
Poisson
Bernoulli
Binomial
What is a uniform distribution?
All events have a equal probability of occurring. Events are mutually exclusive and collectively exhaustive (the dice).
PMF: 1/n
Describe the poisson distribution
It’s used for count variables and supported in discrete steps of 1 from 0 to infinity. It has one parameter, my, and my > 0.
The poisson is approximately normally distributed for high values of my.
What is a feature of the PMF and the mean
The probability mass is distributed equally on either side of the mean. If the mean is close to zero, then a big amount of the probability mass must be between 0 and 1.
Describe the Bernoulli distribution
Single draw of either succes or failure supported {0,1} with one parameter p (probability).
1 = p
0 = 1 - p
What is the binomial distribution?
Repeating sessions of the Bernoulli. Two parameters:
n = number of trials (discrete from 0 to infinity)
p = probability of n successes (0,1)
Approximation to the normal distribution is possible.
