> Home > Statistics Notes > Probability > Transformed Random Variables
Transformed random variables
Suppose that I have a binomial random variable X, which
as always tells us the probability that we will obtain X successes
in N independent Bernoulli trials. For instance, if the number
of trials is 3 and the success probability per trial is 0.6, then
we can use the Binomial formula to calculate what the probability of
getting 0, 1, 2, or 3 successes is. These are the only possible outcomes
of the experiment; the sample space is {0,1,2,3}.
Now suppose that I transform this random variable
in some way. For instance, suppose that I square it. Now I have
a new random variable, Y. Whenever Xis 0, Yis
0; whenever X is 1, Y is 1; whenever X is 2, Y is 4, and whenever X is
3, Y is 9. This goes the other way too: whenever Y is 9, X is 3;
whenever Y is 4, X is 2; whenever Y is 1, X is 1, and whenever Y is
0, X is 0. The sample space of Y is {0,1,4,9}, and for every element
in the sample space of X, there is exactly one element in the sample space
of Y. If we know the value of X, then we also know the value of Y;
if we know the value of Y, we know the value of X.
We know the probability distribution of X.
What can we say about the probability distribution of Y? Note that
the event "X=0" happens when, and only when, "Y=0", so the probability
that Y=0 must be the same as the probability that X=0. Similarly,
the event X=1 happens when and only when Y=1, so the probability that Y=1
must be the same as the probability that X=1. And the event X=2 happens
when and only when Y=4, so the probability that Y=4 must be the same as
the probability that X=2. And finally, X=3 happens when and only
when Y=9, so the probability that Y=9 must be the same as the probability
that X=3.
But we know the probability that X=0, X=1, X=2,
and X=3. These came from the Binomial formula. And so we know
the probability that Y=0, Y=1, Y=4, and Y=9. The notation P(X=x)
represents the probability that the random variable X will take the particular
value represented by x. For instance, P(X=2) would represent the
probability that X would equal 2. We can summarize this information
about the new random variable Y and its relation to the random variable
X in this table:
