Let x be a continous random variable with probability density function given by  Fx=2x, for o<x <20 otherwise

To clarify, the probability density function (PDF) \( f(x) \) of a continuous random variable \( X \) is given by:

\[
f(x) = \begin{cases}
2x, & \text{for } 0 < x < 20, \\
0, & \text{otherwise}.
\end{cases}
\]

Now, let's analyze the function and determine if it satisfies the properties of a valid probability density function.

### 1. **Non-negativity**:
The function \( f(x) \) is non-negative for all \( x \) in its domain, which is true here since \( 2x \) is non-negative for \( x > 0 \).

### 2. **Normalization**:
For \( f(x) \) to be a valid probability density function, the total area under the curve must equal 1. This means we need to check the integral of \( f(x) \) over its entire domain \( 0 < x < 20 \):

\[
\int_{0}^{20} 2x \, dx
\]

Let's compute this integral:

\[
\int_{0}^{20} 2x \, dx = 2 \int_{0}^{20} x \, dx
\]

The integral of \( x \) with respect to \( x \) is:

\[
\int x \, dx = \frac{x^2}{2}
\]

So, we have:

\[
2 \int_{0}^{20} x \, dx = 2 \left[ \frac{x^2}{2} \right]_{0}^{20} = \left[ x^2 \right]_{0}^{20} = 20^2 - 0^2 = 400
\]

Since the integral evaluates to 400, the function \( f(x) = 2x \) over the interval \( 0 < x < 20 \) does **not** satisfy the normalization condition for a probability density function, because it should integrate to 1, not 400.

### **Normalization of the PDF**:
To normalize the function, we must divide the function by the integral of \( f(x) \) over \( 0 < x < 20 \):

\[
f_{\text{normalized}}(x) = \frac{1}{400} \cdot 2x = \frac{1}{200}x
\]

So the valid PDF should be:

\[
f(x) = \begin{cases}
\frac{1}{200}x, & \text{for } 0 < x < 20, \\
0, & \text{otherwise}.
\end{cases}
\]

Would you like to go into more detail about the implications of this PDF or have any other questions?

Here are some related questions you might find useful:
1. What is the expected value of \( X \) given the normalized PDF?
2. How would you compute the variance of \( X \) from this PDF?
3. Can you find the median of the distribution?
4. What is the cumulative distribution function (CDF) corresponding to this PDF?
5. How would you find the probability that \( X \) lies between 5 and 15?
6. What is the mode of the distribution, if it exists?
7. How does the skewness of the distribution look?
8. Can you find a confidence interval for the mean?

**Tip**: Normalizing a probability density function is crucial to ensure that the total probability across the entire domain equals 1, which is a fundamental property of all PDFs.

Explore the concept of probability density functions (PDFs) with a focus on normalization. Learn how to analyze and normalize a given PDF using an example. Suitable for university-level students and anyone interested in probability theory.

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