NAME:
___________________________
1 of 4 pages
4/30/2007
STAT 427
Midterm I
Spring, 2007
Prof. Goel
MWF 9:30  10.20
REMARKS:
§
This is a closed book examination. However, you are allowed to use your own
calculator as well as two (8.5” x 11” twosided) cribsheets.
§
There are four problems worth a total of 60 points.
§
Please write your name on all four sheets.
§
You must explain each step in your answer to get partial credits. Good Luck!
Problem 1. [15 points]
The probability density function of the time
X
(in minutes) that a flight
from Columbus to Detroit arrives earlier or later than its scheduled arrival is given by
f
(
x
) =
≤
≤

.
0
,
6
6
for
)
x

(36
2
elsewhere
x
c
The negative values of
x
indicate flight arriving early, while positive values of
x
indicate flight
arriving late.
(a)
[5 points]
Find the constant c in the above density.
Note that
f(x) = f(x
), therefore the above
pdf
is symmetric around zero. Now
6
66
3
22

60
0
33
(36

)
2

2 (3
6
)
3
12
2 [(36*
6
6
)
0
] 2 *6
*
288
Now 28
8
1
1/288.
x
c
x
d
x
c
x
d
xcx
c
cc
==
=


=⇒=
∫∫
(b)
[4 points]
Find the Expected value of X.
[
]
( ) .
E
X
x
fx dx
=
∫
However, since the
pdf
is symmetric around 0,
E[X]
= 0.
(c)
[6 Points]
Find the probability that one of these flights will arrive between 1 to 3
minutes earlier than its scheduled arrival.
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 Spring '08
 Staff
 Probability, Probability theory, Twin, Prof. Goel

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