Question

Problems
1. Two independent random variables X and Y
have the probability distributions as follows:

X 1 2 5
P (X) 0.2 0.5 0.3

Y 2 4
P (Y) 0.7 0.3

a) Let T = X + Y. Find all possible values of T.
Compute μ and . T σ T
b) Let U = X - Y. Find all possible values of U.
Compute μ U and σ U .
c) Show that μ T

= μ X + μ Y

d) Show that μ U

= μ X − μ Y

e) Show that σ X±Y

=/ σ X ± σ Y

f) Show that σ
2
T
= σ
2
U
= σ
2
X + σ
2
Y

2) Let X be a random variable. Prove that the sum of all deviations from E(X) is always zero. Hints: use the properties of P(X) and basic algebra.

3) Let X be a discrete random variable. Prove that sigma (X) can also be written as sigma (X) = Sqrt ( sum [x^2 times P(x)] - E(x))

Answer 1