Question

Let x be a continuous random variable with the probability density function given by f(x) = 1 if 0 < x < 1, 0 otherwise. Find the probability density function of y = eˣ.

Answer 1

To find the probability density function of y = eˣ, use the transformation technique. Define Y = eˣ and find the cumulative distribution function of Y. Differentiate the cumulative distribution function to find the probability density function of Y.

Step-by-step explanation:

To find the probability density function of y = eˣ, we need to use the transformation technique. Let Y = eˣ, then x = ln(Y). Now we can find the probability density function of Y by finding the derivative of the cumulative distribution function of Y. Since X is a continuous random variable with a probability density function of f(x) = 1 if 0 < x < 1, 0 otherwise, the cumulative distribution function of X is given by P(X ≤ x) = x for 0 < x < 1. Substituting x = ln(Y), we get P(eˣ ≤ Y) = ln(Y) for eˣ < Y < e. To find the probability density function of Y, we differentiate the cumulative distribution function with respect to Y, which gives us f(Y) = 1/Y for eˣ < Y < e.