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Practice Problem: PDF for a linear function of a random variable
Let X be a continuous random variable with pdf $ f_X(x) $. Let $ Y=aX+b $ for some real valued constants a,b, with $ a\neq 0 $. What is the pdf of the random variable Y?
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Answer 1
$ f_{Y}(y)= af_{X}(x)+b $
- Unfortunately, this answer is incorrect. Please try again. This problem is important to help you relate CDF to pdf. Please make sure you understand this well.
- Hint:
- You can start with the definition of CDF with respect to Y, i.e,
- $ F_{Y}(y)= P({Y \leq y}) = P({aX+b \leq y})=... $.
- Use derivative to get pdf of Y since you have CDF of Y.
- make sure to compare two cases for a>0 and a<0.
- -TA
Answer 2
$ F_{Y}(y)= P({Y \leq y}) $.We can define an event $ A={aX+b \leq y}=\left\{ begin{array}{x \leq \frac{y-b}{a}} : a > 0 \\ {x \geq \frac{y-b}{a}} : a < 0 \end{array} \right. $
Answer 3
Write it here.
