4.13 It is known that 15% of US home mortgages are under water (ie, the homeowne

Question

4.13 It is known that 15% of US home mortgages are under water (ie, the homeowner owes more than the house is worth). Suppose 18 mortgages are randomly selected (assume independence). Let the random variable X equal the number that are under water. (a) What is the distribution of X? Be sure to state all parameters. (b) Find the probability that exactly 3 are under water. (c) Find the probability that 1 or more are under water. (d) Given that 1 or more are under water, determine the probability that 2 or more are underwater (e) On average, how many do we expect to be under water? (f) Find Var(x).

Explanation / Answer

a) X is binomial distribution.

p = 0.15

n = 18

b) P(X = x) = 18Cx * 0.15x * (1 - 0.15)18-x

P(X = 3) = 18C3 * 0.153 * 0.8515 = 0.2406

c) P(X > 1) = 1 - P(X = 0)

= 1 - 18C0 * 0.150 * 0.8518

= 1 - 0.0536

= 0.9464

d) P(X > 2 | X > 1) = P(X > 2) / P(X > 1)

= 0.7759 / 0.9464

= 0.8198

e) E(X) = n * p = 18 * 0.15 = 2.7

f) Var(X) = n * p * (1 - p) = 18 * 0.15 * 0.85 = 2.295

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