https://file.aiquickdraw.com/mathbot/file/1729794611544sicv8n4t.jpg

Let's analyze and solve the given problem.

### (a) Explain why the function is a probability density function (PDF)
To be a probability density function, a function must satisfy two conditions:
1. \( f(x) \geq 0 \) for all \( x \).
2. The total area under the curve (the integral of the function over its domain) must equal 1.

From the graph, we can see that:
- The function \( f(x) \geq 0 \) for all values of \( x \) between 0 and 10.
- The total area under the graph (a triangle) must be equal to 1.

Thus, the correct explanation is: 
**"The total area under the graph of the function is 1."**

### (b) Find the probabilities
To compute the probabilities, we need to calculate the area under the curve for the respective intervals.

1. **\( P(X < 3) \)**
   The graph forms a triangle from 0 to 3, and the area under the curve up to 3 gives the probability. The height at \( x = 3 \) is \( 0.15 \).
   The area of this triangle is:
   \[
   A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 0.15 = 0.225
   \]
   So, \( P(X < 3) = 0.225 \).

2. **\( P(3 \leq X \leq 6) \)**
   The interval from 3 to 6 is also a triangle, but its base is 3 (from \( x = 3 \) to \( x = 6 \)) and its height at \( x = 3 \) is 0.15 and at \( x = 6 \) is 0.2. We calculate the area of this trapezoid:
   \[
   A = \frac{1}{2} \times (0.15 + 0.2) \times 3 = \frac{1}{2} \times 0.35 \times 3 = 0.525
   \]
   So, \( P(3 \leq X \leq 6) = 0.525 \).

### (c) Calculate the mean
For a probability density function, the mean \( \mu \) is calculated as:
\[
\mu = \int x f(x) dx
\]
Since the function is piecewise linear (triangles), we calculate it by dividing the area into two triangles and finding the weighted average based on the areas and their respective centroids. However, this requires performing exact integration over the domain of the function. Let me compute that step-by-step for accuracy:

1. **For the first interval (0 to 5)**: We have a triangle whose area we already calculated.
2. **For the second interval (5 to 10)**:

Consider the function whose graph is shown. (a) Explain why the function is a probability density function. (b) Use the graph to find the probabilities P(X < 3) and P(3 ≤ X ≤ 6). (c) Calculate the mean.

Learn how to interpret and calculate probabilities using a probability density function (PDF) from a given graph. Solve for probabilities P(X < 3), P(3 ≤ X ≤ 6), and the mean, using geometry and integration techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation