2.1.1 Probability Functions (PMFs and PDFs) Flashcards
(8 cards)
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.