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The problem you provided involves understanding and working with a probability density function (PDF) \( f(x) \). Let's break down and solve each part:

### Given:
- \( f(x) \) is the probability density function for the weight \( x \) of a female college student, measured in pounds.

### (a) What is the meaning of the integral \( \int_{0}^{\infty} f(x) \, dx \)?

**Answer:**

The integral \( \int_{0}^{\infty} f(x) \, dx \) represents the total probability that a female college student’s weight falls within the range from 0 to \( \infty \) pounds. 

Since \( f(x) \) is a probability density function, this integral must equal 1:

\[
\int_{0}^{\infty} f(x) \, dx = 1
\]

This equation states that the probability of all possible weights for a female college student (from 0 to \( \infty \) pounds) is 1, which makes sense because the weight of a student must fall somewhere in that range.

### (b) Write an expression for the mean of this density function.

**Answer:**

The mean (or expected value) of the weight is given by the following integral:

\[
\mu = \int_{0}^{\infty} x \cdot f(x) \, dx
\]

Here, \( x \cdot f(x) \) represents the product of the weight \( x \) and its probability density at that weight. The integral sums this product over all possible weights to compute the mean.

### (c) How can we find the median of this density function?

**Answer:**

The median \( m \) is the value of \( x \) such that the probability of the weight being less than or equal to \( m \) is 0.5. Mathematically, this is given by:

\[
\int_{0}^{m} f(x) \, dx = 0.5
\]

To find the median, you would solve this equation for \( m \). This equation ensures that 50% of the probability mass lies to the left of \( m \), which is the definition of the median.

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Would you like any additional details or further explanations on any of these parts?

### Related Questions:
1. How do you interpret the variance of a probability density function?
2. What is the difference between mean and median in the context of probability distributions?
3. How would you calculate the standard deviation of this density function?
4. What is the cumulative distribution function (CDF) corresponding to \( f(x) \)?
5. How do skewness and kurtosis describe the shape of a probability distribution?

### Tip:
Understanding the relationship between the PDF, mean, and median of a distribution can help you make informed inferences about the data being modeled.

Suppose f(x) is the probability density function for weight of a female college student, where x is measured in pounds. (a) What is the meaning of the integral ∫0∞ f(x) dx? (b) Write an expression for the mean of this density function. (c) How can we find the median of this density function?

Learn about the probability density function (PDF) for the weight of a female college student. This problem covers key concepts including the meaning of integrals in PDF, calculating the mean and median. Ideal for students in Grades 11-12.

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