Random Variables and Bayes Theorem

Question 1

Bayes theorem, what does it allow?

Formulas

Answer 1

It allows to update the probability for an event based on new evidence or information.

P(H |D) * P(D) = P(D |H) * P(H)

Question 2

random variables can be discrete or continuous, what is the difference?

Answer 2

• A random variable is discrete if the set of possible values can be written as a finite or infinite sequence.
• it is continuous if it takes a continuum of possible values.

Question 3

What is the probability mass function PMF?

How is it indicated?

What is the summation of p(xi)?
What values can p(xi) be?

Answer 3

Given a discrete random variable X, the PMF is the probability of each possible outcome xi.

p(x)=P(X=xi)

• summation of p(xi) = 1
• p(xi) between 0 and 1.

Question 4

What is the probability density function PDF?

How is it indicated? and what is P(a<=x<=b)?

What is f(xi)?
What is the integral from -infinite to infinite of fx(x)?

Answer 4

Given a continuous random variable X, the PDF gives the probability density at each possible x.

f(x),
P(a<=x<=b) is the area under the curve fx(x)

• f(xi) = 0
• integral -inf +inf of fx(x) is 1

Question 5

Given a random variable X,
how is it calculated the expected value of X: E[X]?

What is the variance? the std dev?

Hos is the variance of X: Var(X) calculated?

Answer 5

If X is discrete:
- E[X] = summation( xi * p(xi))
- Var(X) = E[(X-mu)^2] = summation( (xi-mu)^2 * p(xi) )

IF X is continuous:
- E[X] = integral -inf +inf x*f(x) dx
- Var(X) = E[(X-mu)^2] = integral -inf +inf ((x-mu)^2 * f(x) dx)