Questions: Problem 8. (5 points) The continuous random variable, X, has the probability density function given below. P(x<7.5)=□ f(x)= - 0.02 x for 0 ≤ x ≤ 10 - 0 elsewhere

Problem 8.
(5 points)
The continuous random variable, X, has the probability density function given below.
P(x<7.5)=□ f(x)=
- 0.02 x for 0 ≤ x ≤ 10
- 0 elsewhere

Transcript text: Problem 8. (5 points) The continuous random variable, X , has the probability density function given below. \[ P(x<7.5)=\square \quad f(x)=\left\{\begin{array}{ll} 0.02 x & \text { for } 0 \leq x \leq 10 \\ 0 & \text { elsewhere } \end{array}\right. \]

Solution

Solution Steps

To find \( P(x < 7.5) \) for the given probability density function (PDF), we need to integrate the PDF from the lower bound (0) to 7.5. This will give us the cumulative probability up to 7.5.

Step 1: Define the Probability Density Function

The probability density function (PDF) for the continuous random variable \( X \) is given by: \[ f(x) = \begin{cases} 0.02x & \text{for } 0 \leq x \leq 10 \\ 0 & \text{elsewhere} \end{cases} \]

Step 2: Set Up the Integral

To find \( P(x < 7.5) \), we need to compute the integral of the PDF from 0 to 7.5: \[ P(x < 7.5) = \int_0^{7.5} f(x) \, dx = \int_0^{7.5} 0.02x \, dx \]

Step 3: Calculate the Integral

Evaluating the integral: \[ \int 0.02x \, dx = 0.01x^2 + C \] Thus, we compute: \[ P(x < 7.5) = \left[ 0.01x^2 \right]_0^{7.5} = 0.01(7.5^2) - 0.01(0^2) = 0.01(56.25) = 0.5625 \]