Topic 1: Stats & Maths Background Flashcards
What is a discrete random variable?
A random variable that can take on a finite, or a countable infinite set of values, with associated propabilities such that:

What are the rules for probability distributions?
- All probabilities lie between 0 and 1.
- The null set is assigned the probability 0. The full set is assigned 1.
- The probability assigned toi an event that is the union of two disjoint events is the sum of the probabilities assigned to those disjoint events.
What is a CDF?
Cumulative distribution function.
Often denoted F(x), the value at any point is the probabilities that P(X <= x) where X is the random variable.
What is a PDF?
A probability density function.
A function such that:

How are PDF’s related to CDF’s?
For the PDF, denoted by f(x). (see below).
This is conditional on the CDF being differentiable.

What would you expect the CDF and PDF graphs for a normal distribution to look like?

How can the following be calculated?

Where F(x) is the CDF of X.

What is the first moment of a random variable?
It’s expectation (the mean of the population).
How can the expectation of a continous random variable be calculated?

What are higher moments of a random variable?
Expectations of the random variable raised to a higher power.
What is the formula for calculating the nth moment of a continous random variable X?
Where f(x) is the pdf of X.

What is a centered, or central moment? How is it calculated for a CRV?
Rather then taking the value of a variable to some power, the difference between that value and the mean is taken to some power.

What is the significance of the second central moment?
It is the variance of the the random variable, denoted

Define the CDF of a bivariate distribution

How could the CDF of x1 and x2 be calculated if they are both statistically independent?

How can conditional probabilities be defined?
P

What is the formula for conditional density?
Where x2 != 0
** denominator should be fX() **

What is a conditional expectation?
E(X1 | x2)
The expected value of X1, given X2 = x2
Or E(X1 | X2),
Which is a function showing the expected value of X1 given some value of X2
What are some of the properties of conditional expectations, E(X1 | X2), where X2 is a random variable, not a particular value.
- Law of Iterated Expectations:
E(E(X1 | X2 ) ) = E(X)
2.
E(X2 | X2) = X2,
E(X22 | X2) = X2
3.
E(X1 h(X2) | X2) = h(X2) E(X1 | X2) for a deterministic function h
4.
From (3), when E(X1 | X2) = 0,
E(X1h(X2)) = 0
Why is it that E(X1h(X2)) = 0 when E(X1 | X2) = 0

Show the result of the following expression:


Show the result of the following Expression:


Show by example how matrix multiplication is conducted.
Look at the rows and columns of the first and second matricies respectively, with each element of C, Cij being the matrix multiplication of row i and column j.
