https://file.aiquickdraw.com/mathbot/file/1730995698287xhou5n19.jpg

To solve this problem, let's go through each part systematically.

### Given Data:
The probability density function (PDF) of a continuous random variable \( X \) is provided as:
\[
f(x) = \frac{3(8x - x^2)}{256}, \quad 0 < x < 8
\]

### Objective:
1. Find the cumulative distribution function (CDF) \( F(x) \).
2. Calculate the probabilities:
   - (a) \( P(X < 2) \)
   - (b) \( P(X < 9) \)
   - (c) \( P(2 < X < 4) \)
   - (d) \( P(X > 6) \)
   - (e) Find \( x \) such that \( P(X < x) = 0.95 \).

### Step-by-Step Solution

#### Step 1: Find the Cumulative Distribution Function (CDF)
To find \( F(x) \), the cumulative distribution function, we integrate \( f(x) \) over the range \( 0 \) to \( x \):
\[
F(x) = \int_{0}^{x} f(t) \, dt = \int_{0}^{x} \frac{3(8t - t^2)}{256} \, dt
\]

1. **Integrate \( f(t) \):**
   Calculate \( \int \frac{3(8t - t^2)}{256} \, dt \).

2. **Evaluate the definite integral from \( 0 \) to \( x \):**
   Substitute the limits of integration.

#### Step 2: Calculate the Probabilities

Once we have the CDF \( F(x) \), we can use it to calculate each probability as follows:

1. **(a) \( P(X < 2) \)**: Calculate \( F(2) \).
2. **(b) \( P(X < 9) \)**: Since the maximum of \( X \) is 8, \( P(X < 9) = P(X < 8) = F(8) = 1 \).
3. **(c) \( P(2 < X < 4) \)**: Calculate \( F(4) - F(2) \).
4. **(d) \( P(X > 6) \)**: Calculate \( 1 - F(6) \).
5. **(e) Find \( x \) such that \( F(x) = 0.95 \)**: Solve for \( x \) in the equation \( F(x) = 0.95 \).

I’ll perform the calculations for these integrals and probabilities now.Here are the results for each part:

1. **CDF \( F(x) \)**:
   \[
   F(x) = \frac{x^2(12 - x)}{256}
   \]

2. Calculated Probabilities:
   - (a) \( P(X < 2) = F(2) = 0.15625 \)
   - (b) \( P(X < 9) = F(8) = 1 \) (since the maximum range of \( X \) is 8, so \( P(X < 9) = 1 \))
   - (c) \( P(2 < X < 4) = F(4) - F(2) = 0.34375 \)
   - (d) \( P(X > 6) = 1 - F(6) = 0.15625 \)

3. **(e)** To find \( x \) such that \( P(X < x) = 0.95 \):
   - The solution to \( F(x) = 0.95 \) yields \( x \approx 6.92 \).

Let me know if you would like a detailed explanation of any step, or if you have further questions!

Here are some related questions for deeper understanding:
1. How is the cumulative distribution function derived from a probability density function?
2. What does it mean when \( P(X < 9) = 1 \) in this context?
3. How can you interpret the probability \( P(X > 6) = 0.15625 \) in terms of the distribution?
4. Why is the solution for \( x \) when \( F(x) = 0.95 \) significant in probability?
5. What are some applications of calculating specific probabilities for continuous distributions?

**Tip:** When integrating probability density functions, always check that the function integrates to 1 over the defined interval, confirming it as a valid PDF.

Bir sürekli rastgele değişken X'in olasılık dağılım fonksiyonu 0 < X < 8 için f(x) = 3(8x - x^2)/256 şeklinde verilmiştir. Kümülatif dağılım fonksiyonunu ve aşağıdaki olasılıkları hesaplayınız: (a) P(X < 2), (b) P(X < 9), (c) P(2 < X < 4), (d) P(X > 6), (e) P(X < x) = 0.95 ise x=?

Solve for cumulative distribution and probabilities of a continuous random variable X with f(x) = 3(8x - x^2)/256 over 0 < X < 8. Includes step-by-step solutions for P(X < 2), P(X < 9), P(2 < X < 4), P(X > 6), and finding x for P(X < x) = 0.95.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation