Part II: Discrete Variables and their Distributions,

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3. A certain disease has an infection rate 1/5 in a rural area of 40 people. A sample of size 10 is taken at random without replacement. Let X be the number of infected people in the sample.

(1) What is the distribution of X? Give the name of the distribution and values of the parameters;

(2) List all the possible values of X;

(3) Find the probability that exactly 2 individuals in the sample are infected by the disease;

(4) Compute the mean and variance 2 of X using the formulas for this distribution;

(5) Explain the formula for ;

(6) Interpret the probability you calculated in (3).

4. Of all airline flight requests received by a certain discount ticket broker, 10% are for international travels and 90% are for domestic flights. Let X be the number of international flight requests of the next 20 customers. Assuming independence of successive requests, answer the following questions.

(1) What is the distribution of X? Give the name of the distribution and values of the parameters;

(2) List all the possible values of X;

(3) Compute the probability that P(X 5);

(4) Compute the mean and variance of X (using formulas).

(5) Interpret the mean you computed in (4).

(6) Compute the probability in (3) using Poisson approximation (ignore the condition np < 5 or nq < 5).

About Binomial and HyperGeometric Distributions:

If trials are independent (or approximately independent), the resulting distribution is Binomial; otherwise, it is HyperGeometric.

For HyperGeometric distribution, the formulas for the mean and variance are

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