Question

Let x be a random variable that takes values from 0 to 9 with equal probability 1=10. find the pmf of the random variable y = x mod (3).

Answer 1

We have $y\in\{0,1,2\}$, and for each possible outcome we can determine the corresponding probability using the fact that $X$ is uniformly distributed among $x\in\{0,1,\ldots,9\}$.

$y=\begin{cases}0&\text{for }x\in\{0,3,6,9\}\\1&\text{for }x\in\{1,4,7\}\\2&\text{for }x\in\{2,5,8\}\end{cases}$

$\implies\mathbb P(Y=y)=\begin{cases}\dfrac4{10}=\dfrac25&\text{for }y=0\\\\\dfrac3{10}&\text{for }y=1\\\\\dfrac3{10}&\text{for }y=2\end{cases}$