Question

Let X be a random variable with p.d.f. f(x) =6x(1-x), 0\(\le\) x \(\le\)1. Determine the value of b such that P(X < b)=P(X > b).
1. \(\frac{1}{2}\)
2. \(\frac{1 \mp \sqrt{3}}{2}\)
3. 0
4. \(\frac{1}{3}\)

• A. \(\frac{1}{2}\)
• B. \(\frac{1 \mp \sqrt{3}}{2}\)
• C. 0
• D. \(\frac{1}{3}\)

Hint:

Always look for symmetry in continuous probability density functions first.
If the function is symmetric, the median is always the midpoint of the interval, saving you from performing tedious cubic integration.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires finding the median \( b \) of a continuous random variable \( X \) defined by its probability density function (PDF) over a given range.

Step 2: Detailed Explanation:
Let us analyze the given continuous probability density function:
\[ f(x) = 6x(1-x) \quad \text{for } 0 \le x \le 1 \] The condition \( P(X b) \) means that the value \( b \) splits the entire probability distribution into two equal halves.
Since the total area under any valid PDF is \( 1 \), this condition simplifies to:
\[ P(X \le b) = \int_{0}^{b} f(x) \, dx = 0.5 \] Let us solve this using two different approaches:
- Approach 1: Using Symmetry (Conceptual and Fast)
Notice the algebraic structure of \( f(x) \):
\[ f(1-x) = 6(1-x)(1 - (1-x)) = 6(1-x)x = f(x) \] This confirms that the probability density function is perfectly symmetric about the midpoint of its domain \( [0, 1] \), which is \( x = 0.5 \).
For any continuous symmetric distribution on a closed interval, the median must lie exactly at the midpoint of symmetry.
Therefore, \( b = 0.5 = 1/2 \).
- Approach 2: Using Integration (Mathematical Proof)
Set up the integral:
\[ \int_{0}^{b} 6x(1-x) \, dx = 0.5 \] \[ 6 \int_{0}^{b} (x - x^2) \, dx = 0.5 \] \[ 6 \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{b} = 0.5 \] \[ 3b^2 - 2b^3 = 0.5 \implies 4b^3 - 6b^2 + 1 = 0 \] Let us test if \( b = 1/2 \) is a root of this cubic equation:
\[ 4\left(\frac{1}{2}\right)^3 - 6\left(\frac{1}{2}\right)^2 + 1 = 4\left(\frac{1}{8}\right) - 6\left(\frac{1}{4}\right) + 1 = \frac{1}{2} - \frac{3}{2} + 1 = -1 + 1 = 0 \] This mathematical proof confirms that \( b = 1/2 \) is indeed the correct median.

Step 3: Final Answer:
The value of \( b \) is \( 1/2 \).
Therefore, the correct choice is Option (A).