2.1.1 Probability Functions (PMFs and PDFs) Flashcards
What is a probability function?
A function that describes how probabilities are linked to the possible values of a random variable.
What are the two types of probability functions?
Probability mass function (PMF) for discrete random variables and probability density function (PDF) for continuous random variables.
What is a probability mass function (PMF)?
A function that gives the probability that a discrete random variable takes on a specific value.
What are the two conditions a valid PMF must satisfy?
- 0 ≤ P(X = x) ≤ 1 for all x.
- The sum of all probabilities must equal 1: ∑ P(X = x) = 1.
How can a PMF be presented?
As a function of x or as a table listing possible values and their probabilities.
Example: If X is a die roll with a given PMF, how do you find P(3 ≤ X ≤ 4)?
Add P(X=3) and P(X=4) using the given PMF.
What is a probability density function (PDF)?
A function that describes the relative likelihood of a continuous random variable taking on a given value.
Why can’t a PDF provide exact probabilities for specific values?
Because a continuous random variable can take an infinite number of values, making the probability of any single value equal to zero.
The area under a vertical line is zero. DUH
How do you find probabilities using a PDF?
By integrating the PDF over a range:
P(a ≤ X ≤ b) = ∫_{a}^{b} f(x) dx.
What are the two conditions a valid PDF must satisfy?
- f(x) ≥ 0 for all x.
- The total area under the PDF must equal 1: ∫_{-∞}^{∞} f(x) dx = 1.
Can a PDF be greater than 1?
Yes, as long as its integral over all possible values equals 1.
What does evaluating a PDF at a specific value tell us?
The relative likelihood of that value occurring, not the probability.
What does it mean if f(x) = 0 for some values of x?
Those values are impossible for the random variable.
Example: If X represents annual rainfall and has a range of [0, 100], what is P(X = 40)?
P(X = 40) = 0 because the probability of any exact value for a continuous variable is zero.
