Question

Biking accidents Accidents on a level, 3 -mile bike path occur uniformly along the length of the path. The figure below displays the density curve that describes the uniform distribution of accidents. Explain why this curve satisfies the two requirements for a density curve.

Answer 1

A density curve must be non-negative and have an area of 1. The uniform distribution for biking accidents satisfies both by being above the axis and the area under the curve being equal to 1, thus meeting the requirements of a density curve.

Step-by-step explanation:

There are two requirements for a density curve. First, the curve must be on or above the horizontal axis for all values of x. This is because probability cannot be negative.

Second, the total area under the curve must equal 1. This is because the total probability of all outcomes must equal 1.

In the case of the uniform distribution of biking accidents along a 3-mile path, the density function (f(x)) that represents the uniform distribution satisfies both conditions:

1. The density function is a horizontal line above the axis, indicating a non-negative probability of accidents occurring at any point along the path.
2. The area under the curve (which is a rectangle in the case of a uniform distribution) equals 1, indicating the total probability of an accident occurring somewhere on the path is 100%.

Therefore, the described curve exhibits the properties necessary for it to be a density curve.

Answer 2

The density curve for biking accidents on the 3-mile bike path satisfies the two requirements for a density curve because it is non-negative over its entire range and integrates to 1 over the interval.

Step-by-step explanation:

A density curve is a mathematical model that describes the distribution of a continuous random variable. In this case, the uniform distribution of biking accidents along the 3-mile bike path is represented by the density curve. The first requirement for a density curve is that it must be non-negative everywhere. In the context of biking accidents, the density curve ensures that the probability of an accident occurring at any point along the path is always non-negative.

The second requirement for a density curve is that the total area under the curve must equal 1. This corresponds to the total probability of an event occurring, which in this scenario is a biking accident on the 3-mile path. Mathematically, the integral of the density curve over its entire range (3 miles in this case) must equal 1.

This normalization ensures that the probability of an accident happening somewhere on the path is 100%. The uniform distribution, represented by the density curve, satisfies both these requirements, making it a valid model for the distribution of biking accidents along the 3-mile path.

In conclusion, the density curve accurately models the distribution of biking accidents on the 3-mile path, meeting the fundamental requirements for a valid density curve by being non-negative everywhere and integrating to 1 over its range.