EECS 126 — MIDTERM #1
1.
Suppose there exists a test for cancer with the following properties. Let
A
= event that the test states that the tested person has cancer.
B
= event that person has cancer.
It is known that
What is the probability that a person has cancer given that the test says so? Is the test a good
one?
2.
A company produces computer chips at the defective rate of 10%. The good chips have much
longer life than defective ones. The lifetime of good chips has the following cdf:
,
while the lifetime of defective chips has the following cdf:
.
Compute the pdf of the lifetime of an arbitrarily picked chip.
3.
Five people want to play a game of two against two. To decide who should be left out, each of
the five people tosses a fair coin.
If after one round of tossing, the result is one Head and 4 Tails, or one Tail and 4 Heads, the
person whose outcome is different from the rest of the group is out. Otherwise, everyone tosses
again.
What is the probability that it will take exactly
n
th round of tossing to decide?
4.
RV has the density function
Now consider . Find its cdf, pdf, mean, variance, and standard deviation.
P B( ) 0.005=
P A B⁄( ) 0.95=
P BcAc
⁄( ) 0.95=
F1x( ) 1ex10⁄–
–( ) x0≥,=
F2x( ) 1ex2⁄–, x0≥–=
X
fxx1
2
---δx( )+
0
x0 1,[ ] ∈
elsewhere
=
Y X 1
2
---–
2
=
