Download Exam 1, 2011

MIME 4100/5100
Due 3/29/2011, 12:30 PM
Exam 1
3/20/2011
This is a take-home exam. Please sign the honor pledge: I have not given or received any
help in this exam. Name _____________________, Signature ____________________
1) (30 points, 10 points each question)
a. The probabilities of events A and B and their intersection are
P( A)  0.2, P( B)  0.3 and P( A  B)  0.05. Find the probability of
P( AC  B C ) where event AC  B C includes all outcomes that do not
belong to A nor B.
S
A
B
b. Find the probability of getting exactly two heads in five independent flips
of a fair coin.
c. Find the probability of getting 3 or more heads in six flips of a biased coin
for which the probability of heads is 0.7.
2) (35 points) a) (25 points) We tested 500 nominally identical electronic circuits and
observed one failure. Test the hypothesis that the probability of failure of a circuit is
0.01. Use a significance level   0.1.
b) (5 points) Is this an one-tailed or two-tailed test?
c) (5 points) Find the probability that you could reject a hypothesis that is actually true in
this test.
Hint: The test statistic for a sample proportion is,
Pˆ ( A)  P ( A)
ˆ ˆ
P ( A)
(1)
where Pˆ ( A) and P( A) are the observed and true proportions, and
Pˆ ( A){1  Pˆ ( A)}
is the standard deviation of the sample proportions.
P ( A)
nA
The statistic in Equation (1) follows the standard normal distribution.
ˆ ˆ

a. 3) (35 points) Write an Excel program that generates random values from a triangular
probability distribution with minimum value a, most likely value m and maximum
value b. Then test the program by generating 100 numbers from a triangular
distribution with a = 2 min, m = 4 min and b = 7 min.
Hint: The PDF and CDF of a triangular distribution are:
 2( x  a )
for a  x  m

(
b

a
)(
m

a
)

 2(b  x)
f X ( x)  
for m  x  b
(
b

a
)(
b

m
)


 0
otherwise

for x  a
 0

( x  a) 2

 (b  a )( m  a ) for a  x  m

(b  x) 2

for m  x  b
1 
F X ( x)   (b  a )(b  m)

for x  b
1




fX(x)
a
mean 
m
abm
3
a 2  b 2  m 2  ab  am  bm
var iance 
18
b
x