A Note on Applied Probability
Ying Nian Wu, UCLA Statistics
Fall quarter, 2016
According to Maxwell, the true logic of this world is in the calculus of probabilities.
The most fundamental physics laws in quantum mechanics are probabilis
STATS 400 HW6
Problem 1: Prove (1) E[E(Y |X)] = E(Y ). (2) Var(Y ) = E[Var(Y |X)] + Var[E(Y |X)]. (3)
Cov(X, Y ) = E[Cov(X, Y |Z)] + Cov([E(X|Z), E(Y |Z)]. Please define these terms first and then
prove the results.
Problem 2: Let Z N(, 2 ). Let X = Z + 1
STATS 400 HW1 Due Tuesday in class
Problem 1 Suppose we flip a fair coin 5 times independently.
(1) What is the sample space?
A: 32 sequences.
(2) Let X be the number of heads. What is the event X = 3? What is P (X = k) for
k = 0, 1, 2, 3, 4, 5?
A: The ev
STATS 400 HW4
Problem 1 Suppose an urn has r red balls and b black balls. We randomly pick a ball, and
then put two balls of the same color back into the urn. We repeat this process.
(1) What is the probability that the second pick is red?
(2) What is the
STATS 200A HW2
Problem 1: For a continuous random variable X f (x), prove
(1) E[h(X) + g(X)] = E[h(X)] + E[g(X)].
(2) Var(X) = E(X 2 ) E(X)2 .
(3) E(aX + b) = aE(X) + b and Var(aX + b) = a2 Var(X).
(4) Let E(X) = and Var(X) = 2 . Let Z = (X )/. Calculate