2.3.2 Mode

Question 1

what is the mode?

Answer 1

The mode(s) of a set of discrete numbers is the number that appears most often in that set.

Question 2

what is the mode of a discrete r.v.?

Answer 2

The mode(s) of a discrete random variable, X, is/are the value(s) where the probability mass function of X is at its maximum.

Question 3

what is the mode of a continuous r.v.?

Answer 3

The mode(s) of a continuous random variable, X, is/are the value(s) at which the probability density function reaches its maximum.

Question 4

what does the term bimodal mean?

Answer 4

there are two modes

Question 5

How do you find the mode for a discrete distribution?

Answer 5

Calculate each value’s probability and pick the one with the highest probability.

Question 6

Why can’t we brute-force find the mode of a continuous distribution like in the discrete case?

Answer 6

Because continuous variables have infinitely many values, so you can’t check them all.

Question 7

How should you find the mode for a continuous variable on an exam with 5 answer choices?

Answer 7

Just plug each choice into the PDF and pick the one with the highest value.

Question 8

In general, how do you calculate the mode of a continuous variable?

Answer 8

(1) Find critical points by taking the derivative and setting it to 0,
(2) Include endpoints of the domain,
(3) Evaluate the PDF at each critical point,
(4) Pick the one with the highest PDF.

Question 9

What are critical points in this context?

Answer 9

Endpoints of the domain and any ( x ) where the derivative of the PDF is 0.

Question 10

What if the PDF has no critical points (derivative never zero)?

Answer 10

Then the function is monotonic:
- If non-increasing → mode is the lower endpoint
- If non-decreasing → mode is the upper endpoint

Question 11

Can a distribution have more than one mode?

Answer 11

Yes, some are multimodal, but most Exam P distributions will be unimodal.