1. Random Variables and Probability Theory Flashcards
In empirical analysis why is there randomness?
Because empirical models can’t capture all relationships
Random variable
Any variable whose value is a real number that can’t be predicted exactly , and can be viewed as the outcome of chance
A function that governs how probabilities are assigned to interval values for a random variable
CDF
CDF for a random variable x gives the probability that c will take a value less than equal to a specified value. It is a monotonically increasing function of the PDF
How are the PDF and CDF related?
The PDF is a derivative of CDF
Joint PDF
Gives the probability of two random variables will fall in a specified interval
What do we call it when we get the PDF of x or y from the joint distribution?
The marginal PDF
Conditional PDF
When we are interested in the distribution of one random variable given the other variable takes a certain value
When will the conditional distributions of two random variables be the same as the corresponding marginal distributions
When the random variables are independent
Statistical independence
One event occurring has no effect on the prob of the other event occurring
How can we check for independence of two variables
If the joint PDF = the product of the marginal PDFs
f(x,y) =f(x)f(y)
Mean
A random variables average value
Variance
Measures the random variables dispersion around the average value
What is the E(x) for a uniformly distributed random interval?
The midpoint of the interval
What is the expected value of a sum of random variables equal to?
The sum of their expected values
E(X+Y) = E(X) + E(Y)
How can variance be worded?
The average squared deviation of x from its mean.
Or
The expected value of x^2 minus the squared expected value of x
What happens to the variance if all values of x are multiplied by a constant a?
The variance is multiplied by a^2
Covariance
Defined as the expected value of the product of their deviation from their individual expected values
How can the covariance be written in terms of expected values?
C(X,Y) = E(XY) -E(X)E(Y)
Correlation
A standardised measure of covariance which provides a unit free measure of the strength of linear association between x and y
How is correlation calculated?
The covariance divided by standard deviation of each variable
Properties of covariance
- Variance of a sum of random variables equals the sum of variances plus two times the covariance V(X+Y) = V(X) +V(Y) +2C(X,Y)
- If X and Y are independent random variables covariance equals zero (by definition of independence)
How can the variance be calculated from the expected value?
Var(Y) = E(Y^2)- (E(Y))^2
