What characterizes the uniform distribution?

What characterizes the uniform distribution?

• A. Variable probability density function
• B. Constant probability density function on the interval [a,b]
• C. Exponential decrease in probability
• D. Concentration of probability at the center

Answer: (B)

The uniform distribution is characterized by a constant probability density function over a specified interval, typically denoted as [a, b]. This means that within this interval, every value has an equal chance of being chosen, leading to a flat line when the probability density function is graphed. In mathematical terms, the probability density function for a continuous uniform distribution is defined as: \[ f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} \] This definition shows that the height of the density function, which represents probability, remains constant (specifically, \( \frac{1}{b-a} \)), reflecting that all points in the interval [a, b] are equally likely. This property is what distinguishes the uniform distribution from other distributions. In contrast, other options describe properties that are not applicable to a uniform distribution, such as variable densities or specific behavior of densities like exponential decrease or concentration of probability.

The uniform distribution is characterized by a constant probability density function over a specified interval, typically denoted as [a, b]. This means that within this interval, every value has an equal chance of being chosen, leading to a flat line when the probability density function is graphed.

In mathematical terms, the probability density function for a continuous uniform distribution is defined as:

[ f(x) = \begin{cases}

\frac{1}{b-a} & \text{for } a \leq x \leq b \

0 & \text{otherwise}

\end{cases} ]

This definition shows that the height of the density function, which represents probability, remains constant (specifically, ( \frac{1}{b-a} )), reflecting that all points in the interval [a, b] are equally likely.

This property is what distinguishes the uniform distribution from other distributions. In contrast, other options describe properties that are not applicable to a uniform distribution, such as variable densities or specific behavior of densities like exponential decrease or concentration of probability.