https://file.aiquickdraw.com/mathbot/file/1727048801189vlfm4930.jpg

To solve this problem, we are given the probability density function (pdf) for \(x\) as:

\[
p(x) = 0.01(15 - x) \quad \text{for} \quad 0 \leq x \leq 10.
\]

We are tasked with finding the **mean** and **median** of \(x\).

### 1. Mean Calculation

The mean of a continuous random variable \(x\) with probability density function \(p(x)\) over a given interval is given by:

\[
\mu = \int_{a}^{b} x \cdot p(x) \, dx
\]

Substituting the values from the given function:

\[
\mu = \int_0^{10} x \cdot 0.01(15 - x) \, dx
\]

This requires solving the integral of \( x(15 - x) \) over the interval \([0, 10]\).

### 2. Median Calculation

The median \(M\) of a continuous random variable is the value of \(x\) such that the cumulative distribution function (CDF) equals 0.5. Mathematically:

\[
\int_0^M p(x) \, dx = 0.5
\]

Substitute \(p(x) = 0.01(15 - x)\) and solve for \(M\):

\[
\int_0^M 0.01(15 - x) \, dx = 0.5
\]

I will now compute both the mean and the median.### Results:

1. **Mean**: The mean of \(x\) is approximately \( \mu = 4.167 \).
2. **Median**: The median of \(x\) is \( M = 3.820 \) (the relevant solution, as the second solution exceeds the interval [0, 10]).

Let me know if you would like further details or explanations.

### 5 Related Questions:
1. How is the cumulative distribution function (CDF) related to the probability density function (PDF)?
2. How can we interpret the median in the context of a continuous random variable?
3. What are other methods of finding the mean of a random variable besides integration?
4. How does the shape of a probability distribution affect the mean and median?
5. Can the mean and median be the same for certain probability distributions?

**Tip**: When calculating integrals involving polynomials, break the problem into smaller parts to simplify the computations.

A quantity x has density function p(x) = 0.01(15 − x) for 0 ≤ x ≤ 10 and p(x) = 0 otherwise. Find the mean and median of x.

Learn how to calculate the mean and median of a continuous random variable given its probability density function (PDF). This problem involves integration and understanding the relationship between the cumulative distribution function (CDF) and the PDF. Suitable for high school students in Grades 11-12.

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