To find the cdf and pdf of the random variable Y = aX, we need to first consider the relationship between their respective probability distributions. Since Y is a linear transformation of X, we can use the following formula to obtain the pdf of Y:
fy(y) = fx(x) / |a| for a > 0
where fx(x) is the pdf of X and |a| is the absolute value of a. Note that if a < 0, the absolute value is necessary because it changes the direction of the transformation.
To find the cdf of Y, we use the definition:
Fy(y) = P(Y ≤ y) = P(aX ≤ y) = P(X ≤ y/a)
Using the cdf of X, we can write:
Fy(y) = P(X ≤ y/a) = Fx(y/a)
Therefore, the cdf of Y is simply the cdf of X evaluated at y/a.
Putting these two equations together, we get:
fy(y) = fx(x) / |a| for a > 0
Fy(y) = Fx(y/a)
Note that when a < 0, we need to adjust the absolute value in the formula for fy(y).