SECTION 6.1 Discrete Random Variables 325
- Suppose that we draw two cards in succession, and without re-
placement, from a standard 52-card deck. Define the random vari-
ablesX 1 andX 2 by setting
X 1 =
1 if the first card drawn is red
0 if the first card drawn is black;similarly,X 2 =
1 if the second card drawn is red
0 if the second card drawn is black.(a) AreX 1 andX 2 independent random variables?
(b) ComputeP(X 1 = 1).
(c) ComputeP(X 2 = 1).- Suppose that we have two dice and letD 1 be the result of rolling
die 1 andD 2 the result of rolling die two. Show that the random
variablesD 1 +D 2 andD 1 are not independent. (This seems pretty
obvious, right?) - We continue the assumptions of the above exercise and define the
new random variableT by setting
T =
1 ifD 1 +D 2 = 7
0 ifD 1 +D 26 = 7.
Show that T and D 1 are independent random variables. (This
takes a bit of work.)- LetX be a discrete random variable and leta be a real number.
Prove that Var(aX) =a^2 Var(X). - LetXbe a constant-valued random variable. Prove that Var(X) =
0. (This is very intuitive, right?) - John and Eric are to play the following game with a fair coin.
John begins by tossing the coin; if the result is heads, he wins and
the game is over. If the result is tails, he hands the coin over to