Suppose that a man leaves for work between 8:00 a.m. and 8:...

Suppose that a man leaves for work between 8:00 a.m. and $8: 30$ a.m. and takes between 40 and 50 minutes to get to the office. Let $X$ denote the time of departure and let $Y$ denote the time of travel. If we assume that these random variables are independent and uniformly distributed, find the probability that he arrives at the office before $9: 00$ a.m.

Key Concepts

Transformation of Random Variables

Transforming random variables, such as by adding them, involves creating a new variable (in this case, the arrival time is the sum of departure time and travel time). Analyzing this transformed variable requires considering how the individual distributions combine, which is key to determining the probability of arriving before a specified time.

Uniform Distribution

Uniform distribution is a probability distribution where all outcomes in a given interval have an equal chance of occurring. In this context, both the departure time and the travel time are modeled by uniform distributions over specified intervals, meaning that each minute within those intervals is equally likely.

Independence of Random Variables

Independence of random variables means that the outcome of one variable does not affect the outcome of another. Here, the departure time and travel time are independent, so the joint probability distribution is simply the product of the individual uniform distributions.

Joint Probability Distribution

The joint probability distribution of multiple random variables provides the probability of events involving all of the variables together. For independent uniform variables, their joint distribution is constant over the rectangular region defined by their respective intervals, which is essential for calculating probabilities associated with their sum.

Geometric Probability and Integration

Geometric probability involves calculating the probability of an event by determining the area (or volume) of the region where the event occurs relative to the total area (or volume) of the sample space. This approach, often using integration over the defined region, simplifies finding the probability that the sum of the random variables meets a certain condition.

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