Grinstead-Snell-Probability · Problem 6.3.15

Sums of Random Variables

Mathematics · expected-value-variance-continuous

Problem

Let

X

be a random variable distributed uniformly over

[0, 20]

. Define a new random variable

Y

Y = \lfloor X \rfloor

(the greatest integer in

X

). Find the expected value of

Y

. Do the same for

Z = \lfloor X + .5 \rfloor

. Compute

E\bigl(|X - Y|\bigr)

and

E\bigl(|X - Z|\bigr)

. (Note that

Y

is the value of

X

rounded off to the nearest smallest integer, while

Z

is the value of

X

rounded off to the nearest integer. Which method of rounding off is better? Why?)