2.4.1 Discrete Uniform

Question 1

What is a discrete uniform distribution?

Answer 1

A probability distribution where a finite number of outcomes are equally likely to occur.

Question 2

What are the parameters of a discrete uniform distribution?

Answer 2

Two values:
- ( a ) (the lowest possible value)
- ( b ) (the highest possible value)

Question 3

How do we write that a random variable ( X ) follows a discrete uniform distribution?

Answer 3

( X sim ext{Uniform}(a, b) ), meaning ( X ) takes equally likely values from ( a ) to ( b ).

Question 4

What is the probability of each outcome in a discrete uniform distribution?

Answer 4

Since all outcomes are equally likely, the probability of any specific value ( x ) is:
[ P(X = x) = \frac{1}{b - a + 1} ]

Question 5

What does the PMF for the discrete uniform tell us about the relationship between the probability and specific values of ( X )?

Answer 5

The probability does not depend on what ( X ) is, as long as it is in the range ( [a, b] ).

Question 6

What is the mean (expected value) of a discrete uniform distribution?

Answer 6

The average of the first and last values:
[ E[X] = \frac{a + b}{2} ]

Question 7

What is the variance of a discrete uniform distribution?

Answer 7