H OMEWORK 1.5
I) Let X be a RV uniformly distributed in the interval [0, L]. An important function of
such a random variable is g = sin. For simplicity, dene a new random variable
2
X .
L
Y = a sin
The function sin
2
x is periodic with period L. Calculate
H OMEWORK 1.4
I) Let X be an RV with CDF FX and probability density fX . Calculate the CDF and density
function of RV Y = X 1 .
Proof. 1) Cumulative probability distribution (CDF): follow the procedure explained in
class
(1) Cumulative probability distrib
H OMEWORK 1.3
I) Show that the following properties of a Cumulative Distribution Function (CDF) follow
from the denition of the CDF and the properties of probability:
(1)
(2)
(3)
(4)
(5)
F () = 0 and F (+) = 1.
F is a non-decreasing function, i.e. if x1 x
H OMEWORK 1.2
I) If events A and B are independent, then events A and B are independent, and events
A and B are also independent .
Proof. By definition,
A and B independent P(A B) = P (A)P (B).
But B = (A B) + A B , so P (B) = P (A B) + P A B , which yiel
Math 110A HW 1.1 Solutions
SECTION 1.1
10. Suppose the set A has n elements where n Z+ .
(a) How many elements does the power set P(A) have?
We need to decide how many subsets A has. If B is a subset of A, then the elements of
A have two possibilities (ei