Question

If \[ F(x)= \begin{cases} 0, & x[2mm] \left(\dfrac{x+7}{a}\right), & -7[2mm] 1, & x>7, \end{cases} \] represents the probability distribution function of a continuous random variable, then \(a=\)

• A. \(-7\)
• B. \(21\)
• C. \(7\)
• D. \(14\)

Hint:

For a cumulative distribution function, \[ \boxed{ F(-\infty)=0, \qquad F(\infty)=1. } \] Use continuity at the boundary points to determine unknown constants.

Answer 1

The Correct Option is D

Step 1: Use the property of a distribution function. For a cumulative distribution function, \[ \boxed{ F(\infty)=1. } \] Also, the function must be continuous.

Step 2: Apply continuity at \(x=7\). For \[ -7<x<7, \] \[ F(x)=\frac{x+7}{a}. \] At \[ x=7, \] \[ \frac{7+7}{a}=1. \] Hence, \[ \frac{14}{a}=1. \] Therefore, \[ a=14. \] Thus, \[ \boxed{14} \] is the correct answer. Hence, \[ \boxed{(D)} \] is the correct answer.