Recitation 21 Problem 3. There is a nice formula for the expected value of a random variable R that takes on only nonnegative integer values Ex(B)=∑Pr(R>k) Proof ∑P(R>)=P(R=1)+Pr(R=2)+P(R=3)+ +Pr(R=2)+Pr(R=3)+ Pr(r= 3) Pr(R>2) Pr(R=1)+2.Pr(R=2)+3·Pr(R=3)+ Suppose we roll 6 fair, independent dice. Let R be the largest number that comes up Use the formula above to compute Ex(r) Solution. The first task is to compute Pr(r>k); that is, the probability that some die is greater than k. Let's switch to computing the probability of the complementary event Pr(R>k)=1-Pr(R≤k) Now Pr(r< k) is the probability that all the dice show numbers in the set (1, k. If k > 6, then this probability is 1. For smaller k, the probability that one die shows a value in this range is k/ 6. Since the dice are independent, the probability that all 6 dice are in this range is(k /6). Thus, we have Ex(B)=∑Pr(R>k) 、16+26+36+4+56+66� � � �� � � �� � � �� � � Recitation 21 5 Problem 3. There is a nice formula for the expected value of a random variable R that takes on only nonnegative integer values: ∞ Ex (R) = Pr (R > k) k=0 Proof. ∞ Pr (R > i) = Pr (R = 1) + Pr (R = 2) + Pr (R = 3) + · · · i=0 Pr(R>0) + Pr (R = 2) + Pr (R = 3) + · · · Pr(R>1) + Pr (R = 3) + · · · Pr(R>2) . . . = Pr (R = 1) + 2 · Pr (R = 2) + 3 · Pr (R = 3) + · · · = Ex (R). Suppose we roll 6 fair, independent dice. Let R be the largest number that comes up. Use the formula above to compute Ex (R). Solution. The first task is to compute Pr (R > k); that is, the probability that some die is greater than k. Let’s switch to computing the probability of the complementary event: Pr (R > k) = 1 − Pr (R ≤ k) Now Pr (R ≤ k) is the probability that all the dice show numbers in the set {1, . . . k}. If k ≥ 6, then this probability is 1. For smaller k, the probability that one die shows a value in this range is k/6. Since the dice are independent, the probability that all 6 dice are in this range is (k/6)6. Thus, we have: ∞ Ex (R) = Pr (R > k) k=0 � � �6 � � � � �6 � � � 6 � 1 2 6 = 1 + 1 − 6 + 1 − 6 + . . . + 1 − 6 16 + 26 + 36 + 46 + 56 + 66 = 7 − 66