Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Jensen, arriving at a bus stop, just misses the bus. Suppose that he decides to walk if the (next) bus takes longer than 5 minutes to arrive. Suppose also that the time in minutes between the arrivals of buses at the bus stop is a continuous random variable with a \(U(4,6)\) distribution. Let \(X\) be the time that Jensen will wait. a. What is the probability that \(X\) is less than \(4 \frac{1}{2}\) (minutes)? b. What is the probability that \(X\) equals 5 (minutes)? c. Is \(X\) a discrete random variable or a continuous random variable?

Step-by-Step Solution

Answer

a. 0.25; b. 0; c. Continuous.

Step 1: Understanding the Distribution

The problem states that the time between bus arrivals follows a uniform distribution, denoted as \(X \sim U(4,6)\). This means that \(X\) is uniformly distributed between 4 and 6, where each value in this interval is equally likely.

Step 2: Find Probability for Less than 4.5 Minutes

For a uniform distribution \(U(a,b)\), the probability that \(X\) is less than a certain value \(c\) that lies within \([a, b]\) is given by the formula \(P(X < c) = \frac{c - a}{b - a}\). Here, \(a = 4\), \(b = 6\), and \(c = 4.5\). Substitute these values into the formula: \[ P(X < 4.5) = \frac{4.5 - 4}{6 - 4} = \frac{0.5}{2} = 0.25 \] Therefore, the probability is 0.25.

Step 3: Find Probability for X Equals 5 Minutes

For a continuous random variable, the probability of it taking any exact value is 0. Thus, for \(X = 5\), \( P(X = 5) = 0 \).

Step 4: Determine the Type of Random Variable

Since \(X\) is described using a uniform distribution on an interval (continuously on a range), and based on the fact that exact probabilities are 0 for specific points, \(X\) is a continuous random variable.

Key Concepts

Uniform Distribution

A uniform distribution is one of the simplest forms of probability distribution and is often considered when all possible outcomes are equally likely. It can be described by two parameters: a starting point, denoted as \(a\), and an ending point, denoted as \(b\). This distribution assumes that every point in the interval [\(a, b\)] is equally likely to occur. Thus, the probability density function (pdf) for a uniform distribution is constant.

If you're observing a uniform distribution between 4 and 6, often written as \(U(4,6)\), this means that any value between 4 and 6 is as likely to be observed as any other value within this range.

• The pdf for this distribution is \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\). Here, \(b - a = 2\), so \(f(x) = 0.5\).
• The uniform distribution is particularly useful when all outcomes are deemed equally probable. In practical terms, Jensen's bus waiting time modeled with a uniform distribution reflects that any given minute within this 2-minute span has the same chance of occurring.

Continuous Random Variable

A continuous random variable is a type of variable that represents outcomes over a continuous range. Unlike discrete random variables, which have distinct and separate values, continuous variables can take any value within a given interval.

• For example, the time until the next bus arrives could be 4.1 minutes, 4.57 minutes, or any fractional value that you can imagine within the interval from 4 to 6 minutes.
• One key property of continuous variables is that the probability of the variable taking any exact single value is always zero because there are infinitely many possible values.

In Jensen's case, the time he waits for the bus (denoted as \(X\)) is a continuous random variable because it is uniformly distributed between 4 and 6 minutes, representing a continuous interval. This indicates that the waiting time could be any fraction of a minute between these two end points.

Probability Calculation

Calculating probabilities for a uniform distribution involves simple arithmetic since it assumes that each outcome within the interval is equally likely. When determining the probability of a continuous random variable like Jensen's bus arrival time, you are often interested in the likelihood that the variable falls within a certain range.

For a uniform distribution \(U(a,b)\), the probability that the variable \(X\) is less than a certain value \(c\) can be calculated using:

• \(P(X < c) = \frac{c-a}{b-a}\), where \(a\leq c\leq b\).
• In the scenario where Jensen wants to know the probability of the bus arriving before 4.5 minutes, we compute it as:

\[ P(X < 4.5) = \frac{4.5 - 4}{6 - 4} = \frac{0.5}{2} = 0.25 \]

This means Jensen has a 25% chance of the bus arriving within 4.5 minutes of his arrival at the bus stop.

Another aspect of probability calculation in continuous distributions is the question of the probability of \(X\) being exactly a specific value, such as 5 minutes. Since \(X\) is continuous, \(P(X = 5) = 0\). This outcome underlines a fundamental property of continuous probabilities: the probability of \(X\) taking an exact value is always zero.