Questions: A random variable follows the continuous uniform distribution between 15 and 45. a. Calculate the following probabilities below for the distribution. 1. P(x ≤ 30) 2. P(x ≤ 40) 3. P(x ≤ 25) 4. P(x=27) b. What are the mean and standard deviation of this distribution? a. 1. P(x ≤ 30)= (Type an integer or decimal rounded to three decimal places as needed.) a. 2. P(x ≤ 40)= (Type an integer or decimal rounded to three decimal places as needed.) a. 3. P(x ≤ 25)= (Type an integer or decimal rounded to three decimal places as needed.) a. 4. P(x=27)= (Type an integer or decimal rounded to three decimal places as needed.) b. The mean of this distribution is . (Type an integer or decimal rounded to two decimal places as needed.)

Solution Steps

To solve the given problem, we need to understand the properties of a continuous uniform distribution. For a continuous uniform distribution between \(a\) and \(b\), the probability density function (PDF) is constant and given by \( \frac{1}{b-a} \). The cumulative distribution function (CDF) is a linear function that increases from 0 to 1 over the interval \([a, b]\).

1. To find \(P(x \leq c)\), we use the CDF formula: \( F(x) = \frac{x - a}{b - a} \).
2. For \(P(x = c)\) in a continuous distribution, the probability is always 0.
3. The mean of a uniform distribution is given by \( \frac{a + b}{2} \).
4. The standard deviation is given by \( \frac{b - a}{\sqrt{12}} \).

The random variable follows a continuous uniform distribution between \(a = 15\) and \(b = 45\).

For a continuous uniform distribution, the CDF is given by: \[ F(x) = \frac{x - a}{b - a} \]

Using the CDF formula: \[ P(x \leq 30) = F(30) = \frac{30 - 15}{45 - 15} = \frac{15}{30} = 0.500 \]

Using the CDF formula: \[ P(x \leq 40) = F(40) = \frac{40 - 15}{45 - 15} = \frac{25}{30} = 0.8333 \]

Using the CDF formula: \[ P(x \leq 25) = F(25) = \frac{25 - 15}{45 - 15} = \frac{10}{30} = 0.3333 \]

For a continuous distribution, the probability of any single point is always 0: \[ P(x = 27) = 0 \]

The mean of a uniform distribution is given by: \[ \text{Mean} = \frac{a + b}{2} = \frac{15 + 45}{2} = 30.00 \]

The standard deviation of a uniform distribution is given by: \[ \text{Standard Deviation} = \frac{b - a}{\sqrt{12}} = \frac{45 - 15}{\sqrt{12}} = \frac{30}{\sqrt{12}} = 8.660 \]

Final Answer