The jurisdiction of a rescue team includes emergencies...

The jurisdiction of a rescue team includes emergencies occurring on a stretch of river that is 4 miles long. Experience has shown that the distance along this stretch, measured in miles from its northernmost point, at which an emergency occurs can be represented by a uniformly distributed random variable over the range 0 to 4 miles. Then, if $X$ denotes the distance (in miles) of an emergency from the northernmost point of this stretch of river, its probability density function is as follows: $$ f(x)= \begin{cases}0.25 & \text { for } 0<x<4 \\ 0 & \text { for all other } x\end{cases} $$ a. Graph the probability density function. b. Find and graph the cumulative distribution function. c. Find the probability that a given emergency arises within 1 mile of the northernmost point of this stretch of river. d. The rescue team's base is at the midpoint of this stretch of river. Find the probability that a given emergency arises more than $1.5$ miles from this base.

The jurisdiction of a rescue team includes emergencies occurring on a stretch of river that is 4 miles long. Experience has shown that the distance along this stretch, measured in miles from its northernmost point, at which an emergency occurs can be represented by a uniformly distributed random variable over the range 0 to 4 miles. Then, if $X$ denotes the distance (in miles) of an emergency from the northernmost point of this stretch of river, its probability density function is as follows:
$$
f(x)= \begin{cases}0.25 & \text { for } 0<x<4 \\ 0 & \text { for all other } x\end{cases}
$$
a. Graph the probability density function.
b. Find and graph the cumulative distribution function.
c. Find the probability that a given emergency arises within 1 mile of the northernmost point of this stretch of river.
d. The rescue team's base is at the midpoint of this stretch of river. Find the probability that a given emergency arises more than $1.5$ miles from this base.

Key Concepts

Uniform Distribution

A uniform distribution models a situation where every outcome within a specified interval is equally likely. It is defined over an interval [a, b] and is characterized by a constant probability density of 1/(b-a) within that interval. This concept is fundamental in probability theory when modeling scenarios with complete randomness and no bias towards any subinterval.

Probability Density Function (PDF) for Continuous Variables

The probability density function (PDF) characterizes the likelihood of a continuous random variable assuming a specific value within its domain. For a uniform distribution, this function is constant over the defined interval, reflecting the fact that all outcomes are equally probable. Graphically, it appears as a horizontal line over the interval, with zero probability outside of it.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a given value. It is derived by integrating the PDF from the lower limit of the distribution up to the point of interest. For a uniform distribution, the CDF is a linear function over the interval, increasing steadily from 0 at the lower bound to 1 at the upper bound.

Probability Computation for Continuous Variables

In continuous probability distributions, the probability of an event occurring within a specific range is determined by integrating the PDF over that range. This approach provides the area under the curve between two points, which directly equates to the probability of the variable falling within that interval.

Transformation of Variables and Reference Points

When problems involve changing the point of reference or adjusting variables (such as finding probabilities relative to a midpoint), it is essential to translate the original variable appropriately. This process ensures that the computed probabilities accurately reflect the new conditions or distances relative to the transformed reference point.

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