Discussion
A continuous random variable \(X\) has the probability density function
\[ f(x) = \begin{cases} \frac{3}{5} e^{-\frac{3}{5}x}, & x \gt 0 \\ 0, & x \le 0. \end{cases} \]
The probability density function of \(Y = 3X + 2\) is
Answer: A
The density of \(Y\) is \(\frac{1}{5}e^{-\frac{1}{5}(y-2)}\) for \(y \gt 2\), and 0 otherwise.
\(X\) is exponential with rate \(3/5\). The map \(y = 3x + 2\) is increasing, with inverse \(x = (y - 2)/3\) and \(dx/dy = 1/3\). By the change of variables formula:
\[ \begin{aligned} f_{Y}(y) &= f_{X}\left(\frac{y-2}{3}\right)\cdot\frac{1}{3} \\ &= \frac{3}{5}e^{-\frac{3}{5}\cdot\frac{y-2}{3}}\cdot\frac{1}{3} \\ &= \frac{1}{5}e^{-\frac{1}{5}(y-2)},\quad y \gt 2. \end{aligned} \]
Since \(x \gt 0\) corresponds to \(y \gt 2\), the density is 0 for \(y \le 2\). As a check, scaling an exponential by 3 divides its rate by 3, from \(3/5\) to \(1/5\). The option keeping rate \(3/5\) forgets the Jacobian, and the rates \(2/5\) and \(4/5\) come from no consistent working.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development