We know that 5% of the people in a certain population have a virus. Suppose that I draw a random sample of 100 individuals: the population is so large in the order of millions) so that, even though I perform the sampling without replacement, my samples may be considered independent of one another (that is, (i) the first and second individuals having the virus are independent events, and (ii) regardless of the first individual, the probability of picking another individual with the virus is still 5%). Let N be the random variable describing the number of individuals, in my sample, with the virus. (a) What is the probability distribution of N? (Do you recognize it? Does it have a special name?)
(b) Compute the expected number E(N) of individuals, in the sample, with the virus.
(c) What is the prob. of getting exactly this many individuals with the virus, in our sample of 100 individuals?
(d) Compute the standard deviation of N. Hint: Use the appropriate formula...