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Suppose that a uniformly distributed random variable X can have each of the seven values

$-1, 0, 1, 2, 3, 4, 5$

and $Y = X^2-2X$

I want to get the probability mass function of Y I used the Transformation of Random Variables and got PMF of y which is $\frac{1}{7\sqrt{1+y}}$

is there another way to get PMF of y without using Transformation of Random Variables?


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up vote 4 down vote accepted

Your answer makes no sense.

Make a table with two columns. The first column is labeled $X$ and has the seven given values listed in the column. The second column is labeled $Y=X^2-2X$ and had the values of $Y$ listed in it; each $Y$ value being the square of the number in the left hand column minus twice the number in the left column.

Now, look at the right hand column. If you have seven different numbers listed there, $Y$ takes on those seven values with probability $\frac{1}{7}$ each. If a number occurs more than once, $Y$ takes on that value with probability $\frac{k}{7}$ where $k$ is how many times that number occurs in the right column.

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