hi! i need help with this problem, please show work so i can follow along! thank

Question

hi! i need help with this problem, please show work so i can follow along! thanks in advance!

4.13 It is known that 15% of US home mortgages are under water (i.e. 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 (e) On average, how many do we expect to be under water? (f) Find Var(X).

Explanation / Answer

a. Converting the problem into statistical terms, there are two outcomes-success (US home mortages under water) and failure (US home mortages not under water), and probability of success, p=0.15. There are n=18 random, independent trials and probability of success is constant throughout the trials. Assume, X be an r.v denoting number of US home mortgages under water. Thus, distribution of X accounts for binomial distribution, P(X,r)=nCr(p)^r(1-p)^n-r, where, r denotes speciifc number of success in n trials, and p is probability of success.

b. P(X=3)=18C3(0.15)^3(1-0.15)^15=0.2406

c. P(X>=1)=1-P(X<1)=1-P(X=0)=1-18C0(0.15)^0(1-0.15)^18=1-0.0536=0.9464

d. P(X>=2)=1-P(X<2)=1-[P(X=0)+P(X=1)]=1-[0.0536+0.1704]=0.776

e. Mean, E(X)=mu=np=18*0.15=2.7

f. Var(X)=sqrt np(1-p)=sqrt 18*0.15*(1-0.15)=1.51

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