https://file.aiquickdraw.com/mathbot/file/1726163635302au933o4j.jpg

The image contains the following probability density function (PDF) of a continuous random variable \( X \):

\[
f(x) = 
\begin{cases} 
k x^n & \text{for } 0 < x < 1, \\
0 & \text{otherwise}.
\end{cases}
\]

Here, \( k \) is a constant, and \( n \) is a parameter.

### To proceed, we can:
1. Normalize this PDF by determining \( k \), using the condition that the total probability must be 1, i.e., \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \).
   
Would you like me to calculate \( k \) and explain the steps? Let me know if you have any other specific questions! 

Here are 5 related questions to consider:
1. How do you find the expected value \( E(X) \) of a continuous random variable with this PDF?
2. How do you compute the variance \( \text{Var}(X) \)?
3. What is the cumulative distribution function (CDF) for this PDF?
4. How does changing \( n \) affect the shape of the distribution?
5. How do you calculate the median of this distribution?

**Tip:** For continuous distributions, always verify that the integral of the PDF over its domain equals 1 (i.e., it’s normalized).

Let X be a continuous random variable with probability density function f(x):
f(x) = { kx^n for 0 < x < 1, 0 otherwise. }

Learn how to normalize the probability density function f(x) = kx^n for a continuous random variable X where 0 < x < 1, and determine the constant k. Explore related concepts like expected value, variance, and how changing n affects the distribution.

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