Question

Suppose that the random variable Y is uniformly distributed on the interval (0, 1). Find the probability density function (pdf) of U = − ln Y using first the Method of Distribution Functions and then the Method of Transformations, verifying all required assumptions. The solution is a named distribution: provide the name and parameters defining this distribution.

Answer 1

To find the pdf of U = -ln(Y), we can use the Method of Distribution Functions or the Method of Transformations. The resulting distribution is the Exponential distribution with parameter λ = 1.

Step-by-step explanation:

To find the probability density function (pdf) of U = -ln(Y), we can use both the Method of Distribution Functions and the Method of Transformations.

Method of Distribution Functions:

1. Start by finding the cumulative distribution function (CDF) of Y.
2. Next, we will apply the transformation U = -ln(Y) to find the CDF of U.
3. Differentiate the CDF of U to find the pdf.
4. Replace Y with the original distribution to obtain the pdf of U.

Method of Transformations:

1. Start with the pdf of Y, which is uniformly distributed on the interval (0, 1).
2. Apply the transformation U = -ln(Y) to find the pdf of U.
3. Simplify the expression and determine the resulting distribution.

The resulting distribution of U is known as the Exponential distribution with parameter λ = 1.