random variable

[logistic_guy]

Let \(\displaystyle X\) be a random variable with probability

\(\displaystyle f(x) =\begin{cases}\frac{1}{3}, & x = 1,2,3\\0 & \ \text{elsewhere}\end{cases} \)

Find the probability distribution of the random variable \(\displaystyle Y = 2X - 1\).

 

[khansaheb]

Let \(\displaystyle X\) be a random variable with probability

\(\displaystyle f(x) =\begin{cases}\frac{1}{3}, & x = 1,2,3\\0 & \ \text{elsewhere}\end{cases} \)

Find the probability distribution of the random variable \(\displaystyle Y = 2X - 1\).

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[logistic_guy]

Let's compute. We have:

\(\displaystyle Y(X) = 2X - 1\)
\(\displaystyle Y(1) = 2(1) - 1 = 1\)
\(\displaystyle Y(2) = 2(2) - 1 = 3\)
\(\displaystyle Y(3) = 2(3) - 1 = 5\)

Solve for \(\displaystyle X\).

\(\displaystyle X = \frac{Y + 1}{2}\)

Then, the probability distribution of \(\displaystyle Y\) is:

\(\displaystyle g(y) = f(x) = f\left(\frac{y + 1}{2}\right) =\begin{cases}\frac{1}{3}, & y = 1,3,5\\0 & \ \text{elsewhere}\end{cases} \)