Suppose a random variable X has a probability density function as shown in img1.jpg c. Find P(X <= 0.4 | X <= 0.8)Note: I already solve part A and the value of k is 6. The function \( f(x) \) is defined as follows:\[f(x) = \begin{cases} kx(1-x) & \text{for } 0 \leq x \leq 1 \\0 & \text{elsewhere}\end{cases}\]This piecewise function describes \( f(x) \) in two parts. For values of \( x \) between 0 and 1 (inclusive), the function is defined as \( kx(1-x) \), where \( k \) is a constant. For values of \( x \) outside this range, the function value is 0. **Transcription for Educational Website:**### Probability Density Function Tasks**a.** Find the value of \( k \) that makes this a probability density function.**b.** Find \( P(0.4 \leq X \leq 1) \).**c.** Find \( P(X \leq 0.4 \mid X \leq 0.8) \).**d.** Find \( F(b) = P(X \leq b) \) and sketch the graph of this function.**Note:** There are no accompanying graphs or diagrams in the provided image.

Answered: Image /qna-images/answer/607209a0-d2de-4257-9020-9d79f49955f5.jpg

Answered: a Find the value of k that makes this a…

a Find the value of k that makes this a probabil- ity density function. b Find P (0.4 < X< 1). c Find P (X< 0.4|X< 0.8). d Find F (b) = P (X< b) and sketch the graph of this function.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Suppose a random variable X has a probability density function as shown in img1.jpg

c. Find P(X <= 0.4 | X <= 0.8)

Note: I already solve part A and the value of k is 6.

The function \( f(x) \) is defined as follows:

\[
f(x) =
\begin{cases}
kx(1-x) & \text{for } 0 \leq x \leq 1 \\
0 & \text{elsewhere}
\end{cases}
\]

This piecewise function describes \( f(x) \) in two parts. For values of \( x \) between 0 and 1 (inclusive), the function is defined as \( kx(1-x) \), where \( k \) is a constant. For values of \( x \) outside this range, the function value is 0.

**Transcription for Educational Website:**

### Probability Density Function Tasks

**a.** Find the value of \( k \) that makes this a probability density function.

**b.** Find \( P(0.4 \leq X \leq 1) \).

**c.** Find \( P(X \leq 0.4 \mid X \leq 0.8) \).

**d.** Find \( F(b) = P(X \leq b) \) and sketch the graph of this function.

**Note:** There are no accompanying graphs or diagrams in the provided image.

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

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