UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering
and Computer Sciences
EECS 126 Fall 1995
Professor Ren
MIDTERM #1
9 October 1995
[20 pts.] 1)
Prove the following statements:
a)
If , then
b)
If , then
c)
d)
For any RV , any ,
[20 pts.] 2)
Box 1 contains 1000 bulbs of which 10% are defective. Box 2 contains 2000 bulbs of which 5% are
defective. Two bulbs are picked from a randomly selected box.
a)
Find the probability that both bulbs are defective.
b)
Assuming both are defective, find the probability that they came from Box 1.
[20 pts.] 3)
Random variable has the density function
Find the cdf, pdf, mean, and variance of .
[20 pts.] 4)
The probability that a driver will have an accident in 1 month is 0.02. Find the probability that he
will have 3 accidents in 100 months.
[20 pts.] 5)
Players #1 and #2 roll dice alternatively starting with Player #1. The player who rolls eleven first
wins. Find the probability that #1 wins.
NOTE:
A Poisson RV has pmf
p A( ) p B( ) p A B∩( )= =
p A BC
∩( ) B AC
∩( )∪( ) 0=
p A( ) p B( ) 1= =
p A B∩( ) 1=
p A B C∩( ) p A B C∩( )( )p B C( )=
X
α0>s0>,
P X α≥( ) esα–E esX
[ ]≤
X
fXx( ) 1
2
--- 1
2
---δx1
4
---–
x1
2
---
≤,+=
X2
Pkαk
k!
------eα– k,0 1…,= =