How to Determine Variance in Rounded Uniform Distributions?

[thebook]

Hi guys,

I am stuck with this problem... Please help!

Define the rounding function [.] to integers as follows:

[x] = j if x ∈ (j-0.5, j+0.5, for j = 0,+-1, +-2,...

Suppose X has a uniform distribution on the interval (a,b). i.e., X~U(a,b) a<b.

1. find a pair a and b such hat Var(X) > Var([X]) and such that VAR(X) < VAR([X])
2. Suppose that a random variable X has a continuous CDF F(x). What are the distributions of F(X) and [F(X)]? and finally R = [F(X)]-F(X)?

 

[Stephen Tashi]

1. find a pair a and b such hat Var(X) > Var([X]) and such that VAR(X) < VAR([X])

Instead of what you wrote, I think the question should say:

1. Find a pair of numbers (a,b) such that Var(X) > Var([X]) and find another pairs of numbers (a,b) such that Var(X) < Var([X]).

 

[haruspex]

2. Suppose that a random variable X has a continuous CDF F(x). What are the distributions of F(X) and [F(X)]? and finally R = [F(X)]-F(X)?

F(X) doesn't mean anything, and neither does 'the distribution of F(x)'. F(x) is the distribution of X.
Should it read:
What are the distributions of [X] and [X]-X?