STT 430 Test 2
Name: ___________________________________________PID:______________________
Solve the following problems by showing all work necessary. No full credit for correct
answers without proper work.
1. The telephone lines serving an airline reservation office are all busy about 60% of the time.
a) If you are calling this office, what is the probability that you will complete your call in
i) the first try
Answer: 0.4
ii) the second try
Answer: 0.6*0.4=0.24
iii) the third try
Answer: 0.6^2*0.4=0.144
b) If you and a friend must complete calls to this office, what is the probability that a total of four
tries will be necessary assuming that your friend will complete the call before you.
Answer: 3*0.6^2*0.4^2=0.1728
2. A particular sale involves four items randomly selected from a large lot that is known to
contain 10% defectives. Let Y denote the number of defectives among the four sold. The
purchaser of the item will return the defectives for repair and the repair cost is given by
C  3Y 2  Y  2. Find the expected repair cost.
Y is binomial,
E(Y) = np = 4*.1 = .4,
Var (Y) = np(1-p) = 4 * .1*.9 = .36
E(Y^2) = Var(Y) + (E(Y))^2 = .36 + .4^2 =.52
E(Y) = 3*.52 + .4 + 2 = 3.96
3. Suppose that a radio contains six transistors, two of which are defectives. Three transistors are
selected at random, removed from the radio, and inspected. Let Y be the number of defective
transistors observed. Find the probability mass function of Y.
N = 6, 2 defectives, select 3. Use hypergeometric distribution for Y = 0, 1, 2.
Y
0
1
2
P(Y)
4/20
12/20
4. Suppose X has the density function f ( x)  k x ( x  1) if 0  x  2 .
a) Find the value of k
DO NOT GRADE #4!!!!
3
1
X  
2
2
b) Find P
c) Find the cumulative distribution function of X
d) Find E(X), the expected value of X.
4/20
5. Let X ~ N (8, 2) , i.e. X has a normal density with mean 8 and standard deviation 2.
a) Find P7  X  9.5
Answer: 0.4648
b) Find P X  9.56 
Answer: 0.2177
c) Find P X  6.8
Answer: 0.2743
d) Find the 95th percentile of X
Answer: 11.29
e) What two values of X capture the middle 80% of the data under this normal distribution?
Answer: 5.437 and 10.563
6) One hour carbon monoxide concentration in air samples from a large city have an
approximately exponential distribution with mean 3.6 parts per million.
a) Find the probability that the carbon monoxide concentration exceeds 9 parts per million during
a randomly selected 1-hour period
λ = 1/36 = 0.278
Answer: 0.0891 = \int_{9}^{\infty} .278*e^{-.278*t}dt
b) Find the probability that the concentration is between 4 and 12 parts per million.
Answer: 0.29332 = \int_{4}^{12} .278*e^{-.278*t}dt