Question

If a random variable is known to be uniformly distributed between two values, specify those values and discuss the characteristics of a uniform distribution.

Answer 1

Uniform distribution involves a random variable that is equally likely to take any value between a lower bound a and an upper bound b. It has a mean of
`(a + b)/2`and a standard deviation calculated as
`-a)² / 12)`. The values within the interval are equally probable with the distribution's density function appearing as a rectangle.

Step-by-step explanation:

Characteristics of Uniform Distribution:

If a random variable is known to be uniformly distributed between two values, we specify those values as a and b, where a is the lower bound, and b is the upper bound of the distribution. In the notation
`X ~ U(a,b)` , the uniform distribution implies that all values x within the interval from a to
`b (a < x < b or a ≤ x ≤ b)` are equally likely to occur. The density function of a continuous uniform distribution has a rectangular shape, and hence it is sometimes referred to as the rectangular distribution.

Uniform distribution is characterized by the following:

The mean (μ) of a uniform distribution is calculated as

`μ = (a + b) / 2.`

The standard deviation, denoted by σ is calculated using the formula

`σ =`
`√((b-a)² / 12).`

Each outcome within the interval is equally probable, and the total area under the distribution's graph (probability density function) is equal to 1 as required by all probability distributions.

When dealing with uniform distribution, it is crucial to note if the distribution is inclusive or exclusive of the endpoints, which affects computations of probabilities and expectations.