Posted: September 17th, 2017

STATISTICS

1. (10 Points) Using the diagram 1, calculate following probabilities. (1 Point of each)
a) Find the value of x
.
b) P(C
| A).
c) P(B
| A.
B).
d) P(B
| A.
B).
e) P(A| A.
B
.
C).
f) P(A.
B
| A.
B).
g) P((A
.
B).).
h) P((A
.
B.
.
C.).
i) P(B
.
(A.
C.)) .
j) P(C
|(A.
B).).

0.04.
0.01.
0.21.
0.1.
0.18.
0.04.
0.07.
x.
0.09.
A
CB 0.05.
S
Diagram 1

2. (10 Points) Suppose that 8 out of 300 chips in a particular box are defective, and

suppose that 3 chips are sampled at random from the box without replacement. The

event A, B, and C are, respectively, the events that the first, second, and third chips

sampled are defective.
a) Draw probability tree with all probabilities. (7 Points)

b) Find the probability of P(A
.
B.
.
C) . (1 Point)
c) Find the probability of P(B
| A.) . (1 Point)
d) Find the probability of P(B.
| A) . (1 Point)

3. (10 Points) If the probabilities of random variable xi
, i
.
1,2,3,4 are given by
xi
10 30 50 80
pi
0.3 0.1 0.4 0.2

a) Plot the probability mass function. (2 Points)

b) Plot the cumulative distribution function. (2 Points)

c) Find the expectation value. (3 Points)

d) Find the standard deviation. (3 Points)

4. (10 Points, g) ~ k) are optional problems) A company that service air conditioner
units in residences and office block is interested in how to schedule its technician in the
most efficient manner. If the random variable X
, taking the values 1, 2, 3, and 4, is the
service time in hours taken at a particular location, and the random variable Y
, taking
the values 1, 2, 3, and 4, is the number of air conditioner units at the location. The
probabilities given by
1 hr 2 hr 3 hr 4 hr
1 unit 0.06 0.03 0.07 0.05
2 units 0.06 0.05 0.1 0.1
3 units 0.04 0.1 0.11 0.03
4 units 0.05 0.06 0.02 0.07

a) Find joint cumulative distribution function. (2 Points)

b) Find the marginal distribution of X
. (1 Point)

c) Find the marginal distribution of Y
. (1 Point)

d) Find the expectation of X
. (2 Points)

e) Find the expectation of Y
. (2 Points)

f) Find the standard deviation of X
. (2 Points)

g) Find the standard deviation of Y
. (2 Points)

h) Find the conditional probability distribution P(X
| Y
.
4) . (3 Points)

i) Calculate the expectation of P(X
| Y
.
4) . (2 Points)

j) Calculate the standard deviation of P(X
| Y
.
4) . (5 Points)

k) Explain the meaning of differences of two results which are d) and i). (3 Points)

5. (10 Points) Suppose that X
~ B(6, 0.4) , the binomial distribution, find
a) Probability mass function and cumulative distribution function. (5 Points)
b) P(X
.
5) . (1 Point)

c) E(X
) . (2 Points)

d) Var(X
) . (2 Points)

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