Unformatted text preview: 18.05. Practice test 1.
(1) Suppose that 10 cards, of which ﬁve are red and ﬁve are green, are
placed at random in 10 envelopes, of which ﬁve are red and ﬁve are green.
Determine the probability that exactly two envelopes will contain a card with
a matching color.
(2) Suppose that a b ox contains one fair coin and one coin with a head
on each side. Suppose that a coin is selected at random and that when it is
tossed three times, a head is obtained three times. Determine the probability
that the coin is the fair coin.
(3) Suppose that either of two instruments might be used for making a
certain measurement. Instrument 1 yields a measurement whose p.d.f. is
f1 (x) = � 2x, 0 < x < 1
0, otherwise Instrument 2 yields a measurement whose p.d.f. is
f2 (x) = � 3x2 , 0 < x < 1
0, otherwise Suppose that one of the two instruments is chosen at random and a mea
surement X is made with it.
(a) Determine the marginal p.d.f. of X .
(b) If X = 1/4 what is the probability that instrument 1 was used?
(4) Let Z b e the rate at which customers are served in a queue. Assume
that Z has p.d.f.
z > 0,
f (z ) =
Find the p.d.f. of average waiting time T = Z .
(5) Suppose that X and Y are independent random variables with the
x > 0,
f (x) =
0, otherwise Determine the joint p.d.f. of the following random variables:
and V = X + Y .
X +Y ...
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