1Solution:
A1
When A1, A2, A3 are disjoint from each other, obviously, A3
A2
P ( A1 A2 A3) = P( A1) + P( A2) + P( A3)
.
When A1, A2 and A3 are not disjoint from each other, A1 A3
P ( A1 A2 A3) < P( A1) + P( A2) + P( A3)
A2
(2) Solution: (
HOMEWORK #1
STATISTICAL INFERENCE
(1) As seen in class Boole's inequality states that
n
P n Ai j=1
i=1
P(Ai ).
Use Venn diagrams to convince yourself that Boole's inequality is true in the case that n = 3. (2) Suppose we conduct an experiment an
2.1 Solution (a) f X ( x) = 42*5(1 - x), 0 < x < 1; y = x^3= g(x), monotone, and y= (0, 1). Use Theorem 2.1.5.
fY ( y ) = f X ( g -1 ( y )
d -1 g ( y ) = 14 y - 14 y 4/3 , 0 < y < 1 =14y - 14y4/3, 0 < y < 1. dy
If one function f(x) is a p.d.f., it
HW2 solutions and hints (1) a. True. P ( E | F ) = P ( E F ) / P ( F ) b. False, when P ( E ) P ( F ) c. True. P ( E ) + P ( F ) = P ( E F ) + P ( E F ) d. False, e.g. P ( E F ) = 0 e. True. E F = E F F (2) b. (.5)(.9) + (.5)(.2) = .55 c. P (
ASSIGNMENT 2
STATISTICS
(1) Suppose that E and F are events. Determine which of the following statements are true and which are false. If the statement is true, explain why; if the statement is false, give a counterexample. (a) P(E F ) P(E|F ) (b)