3. For patients who wanted to use cephalosporin drugs in a hospital, the rapid allergy skin test was used for the cephalosporin drug allergy test. The results showed that the positive rate was 8%. Assume that 20 patients had the allergy test.

(a) What is the distribution of the number of positive patients? What are the parameters that determine its distribution?

(b) Determine the expected value and variance of the number of positive patients.

(c) Determine the pmf of the number of positive patients.

(d) What is the probability of two positive patients?

(e) Determine the cdf of the number of positive patients.

(f) What is the probability that at most one patient is positive?

(g) Describe the graphical characteristics of the probability distribution of the number of positive patients. Assume that the number of patients in the drug allergy test is changed to 100. How does the distribution graph change?

(h) Assume that another type of drug allergy test shows that 1% of patients are positive. After 100 patients have the allergy test, which distribution can be used to approximate the probability distribution of the number of positive patients? Explain the relationship between the two distributions.