2.1.1 Probability Functions (PMFs and PDFs)

Question 1

What is a probability density function (PDF)?

Answer 1

A function that describes the relative likelihood of a continuous random variable taking on a given value.

Question 2

Why can’t a PDF provide exact probabilities for specific values?

Answer 2

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

Question 3

How do you find probabilities using a PDF?

Answer 3

By integrating the PDF over a range:
P(a ≤ X ≤ b) = ∫_{a}^{b} f(x) dx.

Question 4

What are the two conditions a valid PDF must satisfy?

Answer 4

1. f(x) ≥ 0 for all x.
2. The total area under the PDF must equal 1: ∫_{-∞}^{∞} f(x) dx = 1.

Question 5

Can a PDF be greater than 1?

Answer 5

Yes, as long as its integral over all possible values equals 1.

Question 6

What does evaluating a PDF at a specific value tell us?

Answer 6

The relative likelihood of that value occurring, not the probability.

Question 7

What does it mean if f(x) = 0 for some values of x?

Answer 7

Those values are impossible for the random variable.

Question 8

Example: If X represents annual rainfall and has a range of [0, 100], what is P(X = 40)?

Answer 8

P(X = 40) = 0 because the probability of any exact value for a continuous variable is zero.