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Consider the random variable $X$ which has CDF $F_X(x)$ and suppose further that the random variable $Y$ is a linear function of $X$, that is, $Y=aX+b$ where $a$ and $b$ are constants. We obtained the CDF $G(y)$ and $Y$ in terms of $F_X(x)$ when $a>0$. Now how do I do this for the general case of $a$ and derive a single formula?

StubbornAtom
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jahnavi punna
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  • Write the event ${Y\leq y}$ in terms of the RV $X$ using the relationship $Y=aX+b$ using basic algebra and then take the probabilities of such events $G(y)=P(Y\leq y) =...$ – Nap D. Lover Nov 02 '19 at 20:40
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    (If it isn’f clear, when $a=0$ $Y$ is a degenerate RV, the constant $b$, and when $a<0$ you will have to recall the rules on inequalities when dividing by negative numbers...) – Nap D. Lover Nov 02 '19 at 20:57

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