Question

Obtain a general formula the probability density function fᵧY

​ of a random variable Y in terms of the probability density function f ₍ₓ₎X of a random variable X when Y=g(X) for g(x)=1−2e−∣ˣ∣
. b) Use the result of part a) to obtain the probability density function fᵧY for the random variable Y when X is a uniform (−2,2) random variable.

Answer 1

To obtain a general formula for the probability density function of a random variable Y in terms of the pdf of a random variable X, we can use the concept of change of variables in probability density functions. For the specific case when X is a uniform (-2,2) random variable, we can substitute the appropriate values into the formula to find the pdf of Y.

Step-by-step explanation:

To obtain a general formula for the probability density function (pdf) of a random variable Y in terms of the pdf of a random variable X, when Y=g(X) for g(x)=1−2e−∣ˣ∣, we can use the concept of change of variables in probability density functions. The formula for this transformation is:

fᵧY(y) = f ₍ₓ₎X(g^(-1)(y)) * |(d/dy)(g^(-1)(y))|

For the specific case when X is a uniform (-2,2) random variable, we can substitute the appropriate values into the formula to find the pdf of Y.

fᵧY(y) = f ₍ₓ₎X(g^(-1)(y)) * |(d/dy)(g^(-1)(y))| = f ₍ₓ₎X(2-2e^(-|y|)) * |-2e^(-|y|)|