Continuous Random Variable PDF with Normalization

Math Problem Statement

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

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

\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).

Drop file here or Click Here to upload

Probability
Continuous Random Variables
Probability Density Functions (PDF)
Normalization

f(x) = { kx^n for 0 < x < 1, 0 otherwise }
Integral of PDF = 1: ∫_{0}^{1} kx^n dx = 1

Normalization Condition: The total area under the probability density function must equal 1.

College/University Level (Probability Theory)

Find the value of k for a Continuous PDF: f_X(x) = ke^(-x) for x >= 0

Solving PDF Problem for f(x) = k e^{-|x|}: Mean, Variance, and Probability

Continuous Random Variable X with PDF x^5 / k Over [0, 4]

Finding Probabilities for Continuous Random Variables with PDF f(x) = k(1 + x)

Find the Value of 'k' for Probability Density Function f(x)