Question

Suppose that a nonnegative r.v. X has a distribution function F(x), and that Y is the rounding error if X is rounded off to the nearest integer below. Show that Y has the distribution function ∑j=0 [F(j+y)−F(j)](0≤y<1)

Answer 1

The rounding error Y, when X is rounded off to the nearest integer below, has a distribution function of ∑j=0 [F(j+y)−F(j)](0≤y<1).

Step-by-step explanation:

In this problem, we are given a nonnegative random variable X with a distribution function F(x), and we want to find the distribution function of the rounding error Y when X is rounded off to the nearest integer below. The rounding error Y can take values between 0 and 1.

To find the distribution function of Y, we need to consider the probabilities associated with each value of Y. Let's break down the steps:

For each nonnegative integer j, the probability that X is between j and j+1 is given by F(j+1) - F(j). This represents the probability of X taking values between j and j+1.

When X is rounded off to the nearest integer below, the rounding error Y can be thought of as the difference between X and its floor function, i.e. Y = X - floor(X). Since Y can take values between 0 and 1, we need to consider the probabilities associated with each value of Y.

For each value of j, the probability that Y is between 0 and 1 is given by [F(j+1) - F(j)] where 0 ≤ Y < 1. This represents the probability of the rounding error Y taking values between 0 and 1 when X is rounded off to the nearest integer below.

Finally, to find the distribution function of Y, we sum up these probabilities for all possible values of j: ∑j=0 [F(j+y) - F(j)] where 0 ≤ y < 1.

So, the distribution function of the rounding error Y is ∑j=0 [F(j+y) - F(j)] where 0 ≤ y < 1.