2.10.1 Continuous Uniform Flashcards
What is a continuous uniform distribution?
A probability distribution where every interval of the same length within the domain is equally likely to occur.
What are the parameters of a continuous 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 continuous uniform distribution?
( X sim U(a, b) ), meaning ( X ) is uniformly distributed between ( a ) and ( b ).
What is the probability density function (PDF) of a continuous uniform random variable?
The height of the distribution is constant:
[ f(x) = \frac{1}{b - a}, \quad \text{for } a \leq x \leq b ]
If ( x ) is outside this range, the probability is zero.
What does the PDF tell us about the likelihood of different values?
Since the PDF is a constant, every value in the range has the same likelihood, meaning it is equally likely to occur.
What is the cumulative distribution function (CDF) of a continuous uniform random variable?
The probability that ( X ) is less than or equal to a value ( x ):
[ F(x) = \frac{x - a}{b - a}, \quad \text{for } a \leq x \leq b ]
If ( x < a ), then ( F(x) = 0 ). If ( x > b ), then ( F(x) = 1 .
How do you find the probability that ( X ) falls within a certain range?
Use the CDF:
[ P(c \leq X \leq d) = \frac{d - c}{b - a} ]
where ( c ) and ( d ) are within ( [a, b] .
What is the mean (expected value) of a continuous uniform distribution?
The midpoint of the interval:
[ E[X] = \frac{a + b}{2} ]
What is the variance of a continuous uniform distribution?
The formula for variance is:
[ \text{Var}(X) = \frac{(b - a)^2}{12} ]
Example: A bus arrives uniformly between 8:00 and 8:10 AM. What is the probability that it arrives before 8:05 AM?
[ P(X \leq 8:05) = \frac{5}{10} = 0.5 ]
If a random variable ( X ) is uniform on [100, 500], what is its expected value?
[ E[X] = \frac{100 + 500}{2} = 300 ]
What is the probability that ( X ) (from [100, 500]) is greater than 400?
[ P(X > 400) = \frac{500 - 400}{500 - 100} = \frac{100}{400} = 0.25 ]
