Problem 27 - Calculus and its applications

Question

A number \(x\) is selected at random from [4,20] . The probability density function for \(x\) is given by \(f(x)=\frac{1}{16}, \quad\) for \(4 \leq x \leq 20\) Find the probability that a number selected is in the subinterval [9,20]

Step-by-Step Solution

Answer

The probability is \(\frac{11}{16}\).

Step 1: Understand the Probability Density Function (PDF)

The function given is a probability density function (PDF). For a PDF, the area under the curve over the defined interval equals 1. Here, the function is constant: \(f(x) = \frac{1}{16}\) for the interval \([4, 20]\).

Step 2: Identify the Subinterval

We are asked to find the probability that the number \(x\) falls within the subinterval \([9, 20]\). This means we need to calculate the area under the PDF from \(x = 9\) to \(x = 20\).

Step 3: Calculate the Length of the Subinterval

The length of the subinterval \([9, 20]\) can be calculated as: \[ \text{Length of subinterval} = 20 - 9 = 11. \]

Step 4: Use the PDF to Calculate the Probability

Since the PDF is uniform (constant) over \([4, 20]\), the probability that \(x\) is in \([9, 20]\) is the height of the PDF multiplied by the length of the subinterval. Thus, we calculate: \[ P(9 \leq x \leq 20) = f(x) \times \text{Length of subinterval} = \frac{1}{16} \times 11 = \frac{11}{16}. \]

Step 5: Conclude the Probability

The final probability that \(x\) is selected from the subinterval \([9, 20]\) is \(\frac{11}{16}\).

Key Concepts

Uniform Distribution

Uniform distribution is a type of probability distribution that has constant probability across all outcomes over a specific range. In our scenario, we consider the selection of a random number from an interval, [4, 20]. This forms a continuous uniform distribution. The value of the probability density function (PDF) is constant over this interval, specifically \(f(x) = \frac{1}{16}\). The height of the distribution is derived from ensuring that the total area under the curve is equal to 1, as it should be for any probability distribution.
For continuous uniform distribution:

• The PDF is constant: This means each value within the interval is equally likely to occur.
• The area under the curve over the defined interval equals 1: The rectangle formed above the interval has a height that makes the total area equal to 1.

In practice, we find the probability of events occurring within certain intervals by calculating the area over those subintervals.

Continuous Probability

Continuous probability differs from discrete probability, where outcomes are distinct and countable, like rolling a die. Instead, continuous probability deals with outcomes that form a continuum. That means any real value within an interval can potentially be an outcome.
In continuous probability:

• Outcomes can span real numbers across a continuous range.
• Probabilities for specific values are not defined. Rather, probability is calculated for intervals of values.

In our example, we use a probability density function (PDF) to describe likelihood. The PDF ensures that probabilities sum up to 1 over the interval \([4, 20]\). As such, we computed the probability for a subinterval, \([9, 20]\), by finding the area under the PDF within this range.

Subinterval Probability

Subinterval probability focuses on finding the likelihood of outcomes falling within a specific section of the interval, rather than across the entire domain. For continuous distributions like the uniform distribution, this involves calculating the precise area under the PDF for the subinterval.
To calculate subinterval probability:

• Identify the exact range, in this case \([9, 20]\).
• Determine the width or length of the subinterval: Here, it's \(20 - 9 = 11\).
• Multiply this length by the height of the PDF at the subinterval: In this example, it's \(\frac{1}{16} \times 11 = \frac{11}{16}\).

This simple calculation helps us understand the probabilistic behavior for specific sections of range, making it easier to analyze and interpret data in various fields.