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1. (10 points) How many even numbers can be formed from the digits 9, 1,4,5,6, and 9 if each digit
can be used only once?
2. (50 points) Let X and Y denote the lengths of life, in years, of two components A and B, respectively,
ş! x2
in an electronic system. If the joint density function of these variables is
64
0<x<1-ycl
EX,Y) Rx hy
0 < x <1.0<x<1-x:
f(x, y) =
elsewe jey.301-4)*84f CX74
3
Rxdy
*
了
1' Jay You Determine the value k; FED ECX) = 86 x 6xci->)dy cy) 3(1-2) ²
(b) Find the marginal distributions, expected values, variances, and covariance of X and Y;
dy= 1 (C) Determine whether X and Y are dependent or independent;
X(d) Find the probability that the length of life of component A is less than that of component B;
X(e) Find the probability that the length of life of component A is greater than one year, given the
ar
length of life of component B is equal to two year.
xcy
1313. (10 points) The probability that a flight departs on time is 0.3; the probability that it arrives on time
is 0.3; and the probability that it departs and arrives on time is 0.1. Find the probability that it arrives
on time, given that it did not depart on time.
ex
oin
4. (20 points) The waiting time, in hours, between successive speeders spotted by a radar unit is a
continuous random variable with cumulative distribution -8x76 /
le
= 1- e
11-e
dx
x ZO;
dv=e
0,
f(x) = f'(X) =
x < 0.
8 e
V= 1 84
(a) Find the probability of waiting less than 10 minutes between successive speeders;
hind the wyerane waiting time between successiye speeders spotted by a radar unit.
013-0il
u=X
-8X
-81
-8%
ge
