Question

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..

Short answer

The probability that the man arrives at work before 9:00 AM is approximately 0.6, or 60%.

Step-by-step solution

Calculate the Total Time

First, the total time taken from leaving for work till reaching the office can be represented as \(Z = X + Y\). Here, 'Z' represents the total time consumed in the journey.

Generate Uniform Distribution for Both Variables

Create a uniform distribution for both X and Y. As both of them are independent and have a continuous uniform distribution, \(f(x) = f(y) = 1/(b - a)\), where, for X (the time of leaving), 'a' is 0 (representing 8:00 AM) and 'b' is 30 (representing 8:30 AM). So, \(f(x) = 1/(30 - 0) = 1/30\), and for Y (the time of travel), 'a' is 40 and 'b' is 50. So, \(f(y) = 1/(50 - 40) = 1/10\).

Calculate the Joint Density of X and Y

Since X and Y are both independent random variables, their joint density is obtained by multiplying their densities, i.e., \(f(x, y) = f(x) * f(y)\). Therefore, \(f(x, y) = (1/30) * (1/10) = 1/300\).

Determine the Needed Conditions

In order to be at the office before 9 AM, Z must be less than or equal to 60, because after 60 minutes from 8:00 AM, it will be 9:00 AM. Therefore, we have to calculate the probability of \(Z \leq 60\).

Calculate the Probability

Calculate the integral under the area of \(f(x, y)\) that satisfies the condition \(Z \leq 60\) by integrating from the lower limits to the upper limits for both X and Y (taking the limits from the boundaries of Z).

Apply the Formula and Calculate the Numbers

Applying the formula and solving the integration, P(Z ≤ 60) = Double Integration of [1/300]*dxdy, where the limits for dx are from 0 to 20 and for dy are from 40 to (60-x). The result is roughly equal to 0.6 or 60%.

Key concepts

Joint Density

In probability theory, joint density pertains to the joint distribution of two or more continuous random variables. For two variables, their joint probability density function (pdf) gives the likelihood that the variables simultaneously fall within a specified range. In the context of the exercise, we are dealing with two random variables: the time of departure (\(X\)) and the time of travel (\(Y\)).

Given that both these events have their own continuous uniform distribution, the joint density function is the product of their individual densities. This multiplication is valid because the variables are independent. This is a crucial aspect that considerably simplifies the computation of the joint density.

The formula for the joint density when dealing with independent uniform random variables is:

• \[f(x, y) = f(x) \times f(y)\]
• In our case, \(f(x, y) = (1/30) \times (1/10) = 1/300\).

This means that the joint density function \(f(x, y)\) provides a constant probability density across the domain of X and Y.

Random Variables

Random variables are a fundamental concept in statistics and probability, representing quantities with uncertain values. They assign numeric values to every outcome in a sample space, allowing us to study and model real-world processes.

In our exercise, the variables are:

• \(X\): Departure time, ranging from 8:00 to 8:30 A.M.
• \(Y\): Travel time, ranging from 40 to 50 minutes.

Both \(X\) and \(Y\) are continuous random variables governed by uniform distributions. This means each value within their respective ranges is equally likely. The uniform distribution aids in simplifying calculations and is useful in scenarios where all outcomes are seen as equally possible.

Understanding random variables in this context allows us to calculate probabilities that depend on these quantities, like the probability of arriving before 9:00 A.M.

Independent Events

In probability, events are termed independent if the occurrence of one does not affect the occurrence of the other. For independent random variables, this implies that knowing the outcome of one gives no information about the other.

The exercise assumes that the time the man leaves (\(X\)) and the duration of his travel (\(Y\)) are independent events. This assumption simplifies computation significantly because it allows the joint density to be calculated by multiplying the individual densities of \(X\) and \(Y\).

This independence leads to:

• The joint distribution formula: \[f(x, y) = f(x) \times f(y)\]
• The practical calculation of probabilities without conditional dependency.

In real life, independence means the time taken by the variables isn't influenced by each other. Recognizing such independence in statistical problems allows for simplified, more efficient probability computations.