Final answer:
To find the pdf of Y, use the theorem that relates the pdf of a function of a random variable to the pdf of the original random variable. In this case, Y = 3X - 4. Substitute g^(-1)(y) = (Y + 4)/3 into the formula for fY(y) and simplify to get fY(y) = exp(-2|((Y + 4)/3)|) * 1/3
Step-by-step explanation:
To find the pdf of Y, we can use a theorem that states: if X is a continuous random variable with pdf fX(x), and Y = g(X) is a function of X, then the pdf of Y is given by fY(y) = fX(g^(-1)(y)) * |(dg^(-1)(y))/dy|.
In this case, we have Y = 3X - 4. To find fY(y), we first need to find g^(-1)(y), which is the inverse function of Y. Since Y = 3X - 4, we can solve for X to get X = (Y + 4)/3. Next, we need to find |(dg^(-1)(y))/dy|, which is the absolute value of the derivative of g^(-1)(y) with respect to y. Taking the derivative of (Y + 4)/3 with respect to y gives us 1/3.
Finally, we substitute g^(-1)(y) = (Y + 4)/3 and |(dg^(-1)(y))/dy| = 1/3 into the formula for fY(y), along with the given pdf of X, which is fX(x) = exp(-2|x|). This gives us:
fY(y) = fX((Y + 4)/3) * 1/3 = exp(-2|((Y + 4)/3)|) * 1/3.