Problem 24 - Calculus Concepts

Question

Let \(w\) with input \(x\) be a uniform density function with \(a=4\) and \(b=20\) a. Write \(W\), the corresponding cumulative distribution function. b. Use both \(w\) and \(W\) to calculate the probability that \(x \leq 5.8\) c. Use \(W\) to calculate the probability that \(15

Step-by-Step Solution

Answer

a. \( W(x) = \frac{x-4}{16} \) for \( 4 \leq x \leq 20 \). b. \( P(x \leq 5.8) = 0.1125 \). c. \( P(15 < x \leq 18) = 0.1875 \).

Step 1: Write the Uniform Density Function

The uniform density function, denoted as \( w(x) \), is defined for a continuous random variable \( x \) over the interval \([a, b]\) as follows:\[ w(x) = \frac{1}{b-a} \text{ for } a \leq x \leq b \]For this exercise, \( a = 4 \) and \( b = 20 \). Hence, the function becomes:\[ w(x) = \frac{1}{20-4} = \frac{1}{16} \text{ for } 4 \leq x \leq 20 \]

Step 2: Write the Cumulative Distribution Function W

The cumulative distribution function \( W(x) \) for a uniform distribution is: \[ W(x) = \begin{cases} 0, & x < a \ \frac{x-a}{b-a}, & a \leq x \leq b \ 1, & x > b \end{cases} \]Substitute \( a = 4 \) and \( b = 20 \) into the function:\[ W(x) = \begin{cases} 0, & x < 4 \ \frac{x-4}{16}, & 4 \leq x \leq 20 \ 1, & x > 20 \end{cases} \]

Step 3: Calculate Probability P(x ≤ 5.8) Using w and W

To find the probability \( P(x \leq 5.8) \), use the cumulative distribution function:\[ W(5.8) = \frac{5.8 - 4}{16} = \frac{1.8}{16} = 0.1125 \]This means the probability \( P(x \leq 5.8) = 0.1125 \).

Step 4: Calculate Probability P(15 < x ≤ 18) Using W

To find \( P(15 < x \leq 18) \), calculate:\[ P(15 < x \leq 18) = W(18) - W(15) \]Calculate \( W(18) \) and \( W(15) \):\[ W(18) = \frac{18 - 4}{16} = \frac{14}{16} = 0.875 \]\[ W(15) = \frac{15 - 4}{16} = \frac{11}{16} = 0.6875 \]Thus, \( P(15 < x \leq 18) = 0.875 - 0.6875 = 0.1875 \).

Key Concepts

Probability Calculation

When you're dealing with probability calculations in uniform distributions, you're exploring how often a certain event is expected to happen. For a uniform distribution, the probability across each interval is constant. For example, in our exercise, we were given the interval \([4, 20]\). Probability calculations can be performed using either the density function or the cumulative distribution function.

To calculate the probability that a variable, like \(x\), will take on a specific value or fall into an interval, utilize the cumulative distribution function (CDF). It provides the probability that a random variable is less than or equal to a particular value. For instance, when finding the probability that \(x \leq 5.8\), you simply plug 5.8 into the CDF, resulting in a probability of 0.1125.

Another example is finding the probability that \(x\) falls within a range, such as \(15 < x \leq 18\). By calculating the CDF for both endpoints and subtracting, you find the probability for the interval: \(0.875 - 0.6875 = 0.1875\). This straightforward approach makes it easy to work with uniform distributions.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF), denoted as \(W(x)\), gives the cumulative probability that a uniform random variable is less than or equal to a value \(x\). Based on the parameters of the distribution, it steps up from 0 to 1 across the interval.

For the uniform distribution over the interval \([a, b]\), the CDF is defined as:

• 0, for \(x < a\)
• \(\frac{x-a}{b-a}\), for \(a \leq x \leq b\)
• 1, for \( x > b\)

In the given exercise with \(a = 4\) and \(b = 20\), the function becomes \(W(x) = \frac{x-4}{16}\) for any \(x\) within \([4, 20]\). Therefore, this function linearly increases over the interval from 4 to 20. For example, plugging in 18 gives \(0.875\), meaning there's an 87.5% chance that \(x\) is less than or equal to 18 in this uniform distribution.

Density Function

The density function, denoted as \(w(x)\), describes the probability of a uniform random variable taking on a particular value within a specific interval. It's represented by a flat line over the interval, reflecting the equal likelihood of the variable across the range.

For a continuous uniform distribution, the density function is defined as \(w(x) = \frac{1}{b-a}\) over the interval \([a, b]\).
In this exercise, with \(a = 4\) and \(b = 20\), the density function is \(w(x) = \frac{1}{16}\). This means every number within the interval from 4 to 20 has the same probability density. It's essential to note, however, that for continuous ranges, the probability of the variable taking any exact value is essentially zero. Instead, probability is considered over intervals, such as "between 15 and 18."

Uniform density functions are particularly straightforward because they don't change value within a given interval, making them an excellent tool for beginners to learn the basics of continuous probability distributions.