Question

If the p. d. f. of a continuous random variable X is given by \(f(x) = \{\begin{array}{cc}k(9+8x-x^2), & \text{for }-1\leq x\leq 4 \\ 0, & \text{otherwise}\end{array}\) then the value of \(k\) is \(\ldots\)

• A. \(\frac{1}{125}\)
• B. \(\frac{3}{250}\)
• C. \(\frac{3}{125}\)
• D. \(\frac{1}{250}\)

Hint:

Total probability must be 1, so integrate the pdf from -1 to 4 and set it equal to 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a continuous random variable, the total area under the pdf equals 1. This fixes the constant \(k\).

Step 2: Key Formula or Approach:
\[ \int_{-1}^{4}k\left(9 + 8x - x^2\right)dx = 1 \]

Step 3: Detailed Explanation:
Find the antiderivative:
\[ \int(9 + 8x - x^2)\,dx = 9x + 4x^2 - \frac{x^3}{3} \]
At \(x = 4\):
\[ 36 + 64 - \frac{64}{3} = 100 - \frac{64}{3} = \frac{236}{3} \]
At \(x = -1\):
\[ -9 + 4 + \frac13 = -5 + \frac13 = -\frac{14}{3} \]
Difference:
\[ \frac{236}{3} + \frac{14}{3} = \frac{250}{3} \]
So \(k\cdot\dfrac{250}{3} = 1\), which gives
\[ k = \frac{3}{250} \]
Option (D) \(\tfrac1{250}\) forgets the factor 3 from the cubic term. Options (A) and (C) are based on the number 125, which would arise from a wrong value of the integral.

Final Answer:
\(k = \dfrac{3}{250}\), option (B). \[ \boxed{\frac{3}{250} \text{ (B)}} \]