In a certain city district, the need for money to buy drugs is the reason for 75
ID: 3130119 • Letter: I
Question
In a certain city district, the need for money to buy drugs is the reason for 75% of all thefts. Also, it is known that 40% of thefts involve a knife. Assume that whether a theft involves a knife is independent of whether the theft is because of the need for money to buy drugs. Suppose for the following questions that we are interested in the next 2 thefts that occur in this city district.
(1) Specify the joint distribution of the two random variables in this problem.
(2) Find the probability that exactly 1 of the next 2 thefts is because of drugs.
(3) Find the probability that at most 1 of the next 2 thefts involves a knife.
(4) Find the expected number of thefts that involve a knife.
(5) Find the standard deviation of the number of thefts that are because of drugs.
(6) What is the covariance between the number of thefts that involve a knife and the number of thefts that are because of the need to buy drugs?
Explanation / Answer
X- theft
Y - with knife theft
Since X and Y are independent pdf of x,y will be
f(x,y) = f(x) g(y)
=0.75(0.40)= 0.30
----------------------------------
2) P(due to drugs =1 in 2 trials)
= 2C2(0.75) (0.25)
= 0.1875
3) P(X<=1) = P(1) +P(0)
=2(0.6)(0.4)+0.60^2
= 0.48+0.36
= 0.84
4) E( thefts inv a knife) = np = 100*0.4 = 40
5) Variance =npq = 40(0.6) = 24
Std dev of drugs = 4.90
6) Covariance =0 since events are independent.