xpected value I Corollary 2. If R is a natural-valued random variable, then B)=∑iPr(R= b There is another way to write the expected value of a random variable that takes on lues only in the natural numbers, N=10, 1, 2,.. Theorem 3. If R is a natural-valued random variable, then Proof. Consider the sum Pr(R=1)+Pr(R=2)+Pr(R=3)+ +Pr(R=2)+Pr(R=3)+ Pr(R=3)+… The columns sum to 1. Pr(R=1), 2. Pr(R=2),3. Pr(R=3), etc. Thus, the whole sum ∑iP(R=i)=Bx(F) Here, were using Corollary 2. On the other hand, the rows sum to Pr(R>O), Pr(R> 1 Pr(R> 2),etc. Thus, the whole sum is also equal to ∑P(R>) These two expressions for the whole sum must be equal, which proves the theorem. D 2.1 Mean Time to failure Lets look at a problem where one of these alternative definitions of expected value is particularly helpful. A computer program crashes at the end of each hour of use with probability p, if it has not crashed already. What is the expected time until the program crashes? If we let R be the number of hours until the crash, then the answer to our problem is Ex(R). This is a natural-valued variable, so we can use the formula Ex(B)=∑Pr(R>i� � � � � Expected Value I 5 Corollary 2. If R is a natural­valued random variable, then: ∞ Ex (R) = i · Pr (R = i) i=0 There is another way to write the expected value of a random variable that takes on values only in the natural numbers, N = {0, 1, 2, . . .}. Theorem 3. If R is a natural­valued random variable, then: ∞ Ex (R) = Pr (R > i) i=0 Proof. Consider the sum: Pr (R = 1) + Pr (R = 2) + Pr (R = 3) + · · · + Pr (R = 2) + Pr (R = 3) + · · · + Pr (R = 3) + · · · + · · · The columns sum to 1 · Pr (R = 1), 2 · Pr (R = 2), 3 · Pr (R = 3), etc. Thus, the whole sum is equal to: ∞ i · Pr (R = i) = Ex (R) i=0 Here, we’re using Corollary 2. On the other hand, the rows sum to Pr (R > 0), Pr (R > 1), Pr (R > 2), etc. Thus, the whole sum is also equal to: ∞ Pr (R > i) i=0 These two expressions for the whole sum must be equal, which proves the theorem. 2.1 Mean Time to Failure Let’s look at a problem where one of these alternative definitions of expected value is particularly helpful. A computer program crashes at the end of each hour of use with probability p, if it has not crashed already. What is the expected time until the program crashes? If we let R be the number of hours until the crash, then the answer to our problem is Ex (R). This is a natural­valued variable, so we can use the formula: ∞ Ex (R) = Pr (R > i) i=0