2.4.1 Discrete Uniform Flashcards
What is a discrete uniform distribution?
A probability distribution where a finite number of outcomes are equally likely to occur.
What is a common example of a discrete uniform distribution?
Rolling a fair die, where each outcome (1 to 6) has an equal probability of occurring.
What are the parameters of a discrete uniform distribution?
Two values:
- ( a ) (the lowest possible value)
- ( b ) (the highest possible value)
How do we write that a random variable ( X ) follows a discrete uniform distribution?
( X sim ext{Uniform}(a, b) ), meaning ( X ) takes equally likely values from ( a ) to ( b ).
What is the probability of each outcome in a discrete uniform distribution?
Since all outcomes are equally likely, the probability of any specific value ( x ) is:
[ P(X = x) = \frac{1}{b - a + 1} ]
What does the PMF tell us about the relationship between the probability and specific values of ( X )?
The probability does not depend on what ( X ) is, as long as it is in the range ( [a, b] ).
What is the mean (expected value) of a discrete uniform distribution?
The average of the first and last values:
[ E[X] = \frac{a + b}{2} ]
What is the variance of a discrete uniform distribution?
The formula for variance is:
[ \text{Var}(X) = \frac{(b - a + 1)^2 - 1}{12} ]
Example: If ( X ) represents the result of rolling a fair 6-sided die, what is its expected value?
The possible values are 1 to 6, so:
[ E[X] = \frac{1 + 6}{2} = 3.5 ]
If ( X ) is the number of accidents per week and follows a uniform distribution from 0 to 4, what is its mean?
[ E[X] = \frac{0 + 4}{2} = 2 ]
If each accident costs $1,000 in damages, what is the mean reimbursement per week?
Since reimbursement is ( 1,000 \times X ), the mean reimbursement is:
[ 1,000 \times E[X] = 1,000 \times 2 = 2,000 ]
