MATH 475 - Spring 2022 – Homework 0B
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MATH 475 - Spring 2022 – Homework 0B
Questions
1. Let X be a random variable and a e 皿. Show Var(aX) = a苎Var(X).
2. Let X > 0 be a r.v. such that E(X) and E ╱ ∶ are finite. Show E ╱ ∶ > .
3. Let X ~ N(0, 1). Show that
E exp(θX) = exp(θ苎 /2),
for θ e 皿, and also that
E exp(aX苎 + θX) = exp ╱ 2(1 θ苎- 2a)、 ,
for a < 1/2.
4. Let X , Y be discrete random variables on a probability space (Ω , r, 贮). (a) Explain why X and Y can be expressed as
X = x乂I│乞 , Y = y乂I│乞 ,
乂 乂
where x乂 , y乂 e 皿 and {A乂} is a partition of Ω into disjoint sets A乂 e r.
(b) Using (a), prove the Cauchy–Schwarz inequality for X and Y , i.e.
|E(XY)| < ←E(|X|苎 )E(|Y |苎 ).
5. Let X and Y be random variables. Show that the correlation coefficient ρ(X, Y) =
satisfies
-1 < ρ(X, Y) < 1
whenever Cov(X, Y), Var(X), and Var(Y) exist, and Var(X), Var(Y) 0.
2022-03-14