A random variable X has the following probability density function
f(x) = 6x (1 - x) for 0 ≤ x ≤ 1.
Visually, the distribution of X is demonstrated below.

Compute P(0.26 < X < 0.49)
0.3442
(Enter your answer to four decimals)
(b) Compute P(0.82 ≤ X ≤ 1)
0.0613

The problem asks to compute two probabilities for a random variable \( X \) with the given probability density function (PDF):

\[
f(x) = 6x(1 - x) \quad \text{for} \quad 0 \leq x \leq 1.
\]

### (a) Compute \( P(0.26 < X < 0.49) \)

The probability \( P(0.26 < X < 0.49) \) is the integral of the PDF from 0.26 to 0.49:

\[
P(0.26 < X < 0.49) = \int_{0.26}^{0.49} 6x(1 - x) \, dx.
\]

### (b) Compute \( P(0.82 \leq X \leq 1) \)

Similarly, the probability \( P(0.82 \leq X \leq 1) \) is the integral of the PDF from 0.82 to 1:

\[
P(0.82 \leq X \leq 1) = \int_{0.82}^{1} 6x(1 - x) \, dx.
\]

Let's calculate both integrals. I'll start with part (a).The results of the integrals are:

(a) \( P(0.26 < X < 0.49) \approx 0.3174 \)

(b) \( P(0.82 \leq X \leq 1) \approx 0.0855 \)

It seems that the values slightly differ from the provided ones. If you'd like, I can explore further to clarify any discrepancies.

Do you want more details or have any questions? 

Here are 5 related questions for further exploration:
1. How do we derive the probability density function for \( X \)?
2. What is the expected value \( E(X) \) for this distribution?
3. How does the cumulative distribution function (CDF) relate to the PDF?
4. Can we compute the variance of \( X \) for this distribution?
5. What is the shape of this distribution, and how does it affect the probabilities?

**Tip:** To compute probabilities from a PDF, always integrate the PDF over the desired range of \( X \).

A random variable X has the following probability density function: f(x) = 6x(1 - x) for 0 ≤ x ≤ 1. Compute P(0.26 < X < 0.49) and P(0.82 ≤ X ≤ 1).

This problem explores computing probabilities for a random variable X with the PDF f(x) = 6x(1 - x) using integration. Detailed solutions include calculating P(0.26 < X < 0.49) and P(0.82 ≤ X ≤ 1), suitable for undergraduate probability students.

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