Question

A photo-sharing startup offers the following service. A client may upload any number N of photos and the server will compare each of the N 2 pairs of photos with their proprietary image matching algorithms to see if there is any person that is in both pictures. Testing shows that the matching algorithm is the slowest part of the service, taking about 100 milliseconds of CPU time per photo pair. Hence, estimating the number of photos uploaded by each client is a key part of sizing their data center. The people in charge say that their gut feeling is that N = 10. You (the chief technical officer) say, “but N is a random variable”. What will the average CPU demand per client (as a function of N, p or λ) if N follows

• the “distribution” where N is the same fixed number with probability 1?

• the Poisson distribution with parameter λ?

• the geometric distribution with parameter p?

• N = 80X + 5, where X is a Bernoulli random variable with parameter p?

In each case, include as part of your answer the expected value of N and the variance of N.

Answer 1

1. the “distribution” where N is the same fixed number with probability 1?

For average CPU demand per client:

Expected value of N = N

Variance of N = 0

2. the Poisson distribution with parameter λ?

For average CPU demand per client:

Expected value of N =

Variance of N =

3. the geometric distribution with parameter p?

For average CPU demand per client:

Expected value of N = 1/p

Variance of N = 1/p

4. N = 80X + 5, where X is a Bernoulli random variable with parameter p?

For average CPU demand per client:

Expected value of N = E[80X + 5] = 80E[X] + 5 = 80*p+5 = 80p+5

Variance of N = 80² Var(X) = 80² * p(1-p) = 6400p(1-p)