Determine whether each random variable described below satis

Determine whether each random variable described below satisfies the conditions for a binomial stting, a geometric setting, or neither. Support your conclusion WITH EVIDENCE in each case.

a. Suppose that one of every 100 people in a large community is infected with HIV. You want to identify an HIV-positive person to include in a study of an experimental new drug. How many individuals would you expect to have to interview in order to find the first person who is HIV-positive?

b. Deal seven cards from a standard deck of 52 cards. Let H = the number of hearts dealt.

Solution

a. Given proportion of people of HIV in the community. That is 1/100

Here we want to find the person whi HIV positive. This experiment stops if we find the HIV positive.

The number of failures (non HIV positive) is a random variable before getting first success (HIV positive) is a random variable.

This follows geometric distribution.

b)

Total cards given = 52; In standard deck there are 13 hearts

Seven cards drawn from 52 cards and the random variable H is the number of hearts dealt

If seven cards drawn with replacement then it follows binomial distribution.

the probability of heart = 1/13 ; number of draws = 7

H is the number of hearts.

so This follows binomial distribution with parameters ( H,7,1/13)

If drawn without replacement, then is neither binomial nor geometric distributions.

Note: In this case

The random variable H can take values of 1,2,....min ( sample,no.of hearts)

The random variable H follows hyper geometric distribution.