Suppose p = 0.5. Recall that Facebook gives Ann a credit of 4 cents per friend request she accepts. Suppose now that LinkedIn gives her a credit of 9 cents per connection request she accepts. Ann accepts all requests from Facebook and LinkedIn. On some days she earns more credit from Facebook, and on some days she earns more credit from LinkedIn. Let the random variable Z denote the amount of credit she receives from LinkedIn minus the amount of credit she receives from Facebook, in cents. Then, what is the expected value of Z?

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BlackTom AI · Accepted Answer

We are given a problem about daily credit earnings from two different platforms and a random variable Z defined as LinkedIn credit minus Facebook credit, in cents. The key pieces are:
- Facebook pays 4 cents per accepted friend request.
- LinkedIn pays 9 cents per accepted connection request.
- p = 0.5 is provided, which typically would be the probability of accepting a given request.
- Ann accepts all requests from both platforms on a day, so the actual credit on a day depends on how many requests she receives and how many of those are accepted (given she accepts all, acceptance is effectively guaranteed for each received request). 
- Z is the difference: Z = (LinkedIn credit) − (Facebook credit).
- We are asked for E[Z], the expected value of Z.

To reason about E[Z], separate the contributions from each platform and use linearity of expectation: E[Z] = E[LinkedIn credit] − E[Facebook credit].

Step-by-step analysis of the components:
- Let L denote the number of LinkedIn connection requests Ann receives on a day. Since she accepts all requests, the number of accepted LinkedIn connections equals L, and the credit from LinkedIn is 9 cents per accepted request. Therefore, LinkedIn daily credit equals 9 × L. Its expected value is E[LinkedIn credit] = 9 × E[L].
- Let F denote the number of Facebook friend requests Ann receives on a day. She also accepts all of these, so the Facebook credit equals 4 cents per accepted request, i.e., 4 × F. Its expected value is E[Facebook credit] = 4 × E[F].
- Consequently, E[Z] = 9 × E[L] − 4 × E[F].

Assuming the processes that generate L and F are governed by the same underlying randomness (for example, similar daily counts of requests with the same distribution) and the problem provides p = 0.5 as the probability of accepting a given request, we would typically relate E[L] and E[F] to the expected number of requests arriving on a day for LinkedIn and Facebook. If the two platforms have the same daily request distribution, then E[L] = E[F], which would imply E[Z] = 9 × E[L] − 4 × E[L] = 5 × E[L], but that simplifies to 5 × E[L], not a fixed number unless E[L] is specified. If E[L] and E[F] are equal due to identical counting processes, E[Z] would be 0 because 9 × E[L] − 4 × E[L] would still yield 5 × E[L], which is nonzero unless E[L] = 0; however, the actual equality E[L] = E[F] does not by itself imply E[Z] = 0 unless the coefficients also balance out in a way that makes the net expectation zero, which they do not here because the per-accept credits are different (9 vs 4).

Another way to view it is to think in terms of the difference in per-accept credits. For each accepted LinkedIn request, you gain 9 cents; for each accepted Facebook request, you lose 4 cents when forming Z. If the expected numbers of accepted requests on a day differ in such a way that the expected positive contributions from LinkedIn exceed the Facebook deductions by 4.4 cents on average, E[Z] would be 4.4. In practice, to obtain a concrete numerical value for E[Z], you would need the expected daily counts E[L] and E[F] (or the joint distribution of L and F). Without those explicit expectations, E[Z] remains expressed as 9E[L] − 4E[F].

Common pitfalls or misconceptions to watch for:
- Assuming E[Z] equals 5 times the difference in expected counts without careful connection to the per-accept credits. The factor 5 arises from the difference in per-accept contributions (9 − 4 = 5), but you still must multiply by the appropriate expected counts of accepted items on a day for each platform.
- Thinking that p = 0.5 automatically makes E[L] = E[F] or cancels the difference. The probability p affects the likelihood of accepting a given request, but the problem statement says Ann accepts all requests, so the acceptance probability does not reduce the counts; instead, the daily counts L and F themselves (which drive E[L], E[F]) determine E[Z] via the 9 and 4 cent rates.
- If one mistakenly assigns identical distributions to L and F and then subtracts, one might incorrectly conclude E[Z] = 0. In reality, unless E[L] and E[F] are calibrated so that 9E[L] equals 4E[F], a nonzero expectation results.

Practical takeaway: the expected value of Z hinges on the relationship between the expected numbers of LinkedIn and Facebook requests Ann receives per day, E[L] and E[F]. The formula E[Z] = 9E[L] − 4E[F] is the correct expression, and any numeric answer requires those two expectations to be specified. If the problem provides a numeric answer like 4.4, that implies a specific relation between E[L] and E[F], namely 9E[L] − 4E[F] = 4.4, which would require additional information about the daily request distributions to verify.

COMM_V 190 101-104 2025W1 FlexHW14.COMM190.2025.W1 (optional — see syllabus)

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