2.10.1 Continuous Uniform

Question 1

What is a continuous uniform distribution?

Answer 1

A probability distribution where every interval of the same length within the domain is equally likely to occur.

Question 2

What are the parameters of a continuous 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 continuous uniform distribution?

Answer 3

( X sim U(a, b) ), meaning ( X ) is uniformly distributed between ( a ) and ( b ).

Question 4

What is the probability density function (PDF) of a continuous uniform random variable?

Answer 4

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.

Question 5

What does the PDF of the continuous uniform tell us about the likelihood of different values?

Answer 5

Since the PDF is a constant, every value in the range has the same likelihood, meaning it is equally likely to occur.

Question 6

What is the cumulative distribution function (CDF) of a continuous uniform random variable?

Answer 6

Question 7

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

Answer 7

The midpoint of the interval:
[ E[X] = \frac{a + b}{2} ]

Question 8

What is the variance of a continuous uniform distribution?

Answer 8

The formula for variance is:
[ \text{Var}(X) = \frac{(b - a)^2}{12} ]

Question 9

If a random variable ( X ) is uniform on [100, 500], what is its expected value?

Answer 9

[ E[X] = \frac{100 + 500}{2} = 300 ]

Question 10

What is the probability that ( X ) (from [100, 500]) is greater than 400 on a continuous uniform?

Answer 10

[ P(X > 400) = \frac{500 - 400}{500 - 100} = \frac{100}{400} = 0.25 ]