- The total interval for \(X\) is \([4.5, 18.5]\), which has a length of \(14\).
- The favorable interval for \(X > 6\) is \([6, 18.5]\), which has a length of \(12.5\).
- The probability is the ratio of the favorable length to the total length: \(\frac{12.5}{14} = \frac{25}{28}\).
- The final probability is \(\boxed{\frac{25}{28}}\).
Explanation
- Understanding the Uniform Distribution
The problem asks us to find the probability \(P(X > 6)\) for a continuous uniform random variable \(X\) that follows the distribution \(X \sim U(a, b)\), where the parameters are \(a = 4.5\) and \(b = 18.5\).
A continuous uniform distribution is defined over an interval \([a, b]\), where every point in that interval has an equal probability density. The probability density function is given by \(f(x) = \frac{1}{b - a}\) for \(a \le x \le b\), and \(0\) otherwise. To find the probability that \(X\) falls within a specific sub-interval, we calculate the length of that sub-interval divided by the length of the total interval \([a, b]\).
- Calculating the Lengths of Intervals
The total length of the interval \([a, b]\) is calculated as:
\[L_{\text{total}} = b - a = 18.5 - 4.5 = 14\]
We are looking for the probability that \(X > 6\). Since the variable \(X\) is constrained to the interval \([4.5, 18.5]\), the condition \(X > 6\) defines the sub-interval \((6, 18.5]\).
The length of this favorable sub-interval is:
\[L_{\text{fav}} = 18.5 - 6 = 12.5\]
- Computing the Probability
The probability \(P(X > x_0)\) for a uniform distribution \(U(a, b)\) where \(a < x_0 < b\) is given by the ratio of the length of the interval \([x_0, b]\) to the total length \([a, b]\):
\[P(X > x_0) = \frac{b - x_0}{b - a}\]
Substituting our values into the formula:
\[P(X > 6) = \frac{18.5 - 6}{18.5 - 4.5} = \frac{12.5}{14}\]
To simplify this fraction, we multiply both the numerator and denominator by \(2\):
\[\frac{12.5 \times 2}{14 \times 2} = \frac{25}{28}\]
Therefore, the probability \(P(X > 6)\) is \(\frac{25}{28}\).
Examples
In real-world applications, uniform distributions model scenarios where an outcome is equally likely to occur within a defined range. For example, if a bus is scheduled to arrive at any time between 4:30 PM (represented as 4.5) and 6:30 PM (represented as 6.5) and arrives uniformly at any time, we could use this logic to find the probability of the bus arriving after 6:00 PM. In this specific case, the math shows that the probability of the event occurring within the remaining duration is the ratio of the favorable window to the total window size.