Expression of Discrete and Continuous random variables.

Question

Let $X$ be a discrete random variable. And let $Y = cX$ for some constant $c$. How can you express the distribution of $Y$ in terms of the distribution of $X$?
Let $X$ be a continuous random variable now. And let $Y = cX$ again. How can you express the PDF of $Y$ in terms of the PDF of $X$ by using CDF?

Keep in mind that if $Y=cX$, then you can also say that $X = c^{-1}Y$, that might help.

Henry · Answer 1 · 2013-08-13 21:30:50Z

Hints:

1) What is the probability that $X=x$? What is the probability that $Y=cx$?

2) What is the probability that $X\le x$? What is the probability that $Y\le cx$? What does this imply for the density?

You might read and inwardly digest Wikipedia's article on scale parameters as well as the answers to your previous question Normal Distribution and PDF

soakley · Answer 2 · 2013-08-25 13:30:22Z

up vote 0 down vote

Let $F(x)$ be the CDF of $X$ and $G(y)$ be the CDF of $Y.$ Then $$G(y)= P[Y \leq y]= P[cX \leq y] = P[X \leq {y \over c}] = F({y \over c}).$$

answered Aug 25 at 13:30

soakley
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2 Answers