Final answer:
The PDF of Y is found by scaling and shifting the PDF of X; the expected value of Y is a times the expected value of X minus b; and the variance of Y is a squared times the variance of X.
Step-by-step explanation:
To find the probability density function (PDF), expected value, and variance of a transformed random variable Y = aX - b, where X is a given random variable and a and b are constants, we use the properties of linear transformations of random variables.
The PDF of Y can be found using the change of variables technique if X is a continuous random variable. Firstly, if X has a PDF fX(x), and Y = aX - b, then the PDF of Y is given by fY(y) = (1/|a|) * fX((y + b) / a). This accounts for the stretching and shifting effect of the linear transformation.
The expected value (mean) of Y is calculated using the linearity of expectation, E[Y] = E[aX - b] = aE[X] - b.
The variance of Y is found by recognizing that variance is affected by scaling but not by shifting, so Var(Y) = a2Var(X).