Question

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The Question:

An FBI survey shows that about 80% of all property crimes go unsolved. Suppose that in your town 3 such crimes are committed and they are each deemed independent of each other. X is the number of crimes will be solved in your town. Complete the table below for the probability mass function and cumulative probability function of the random variable X using the probabilities listed below.

X	0	1	2	3
P(x)	0.512			0.008
F(x)				1

1. (5 points) The values missing from the table are:

2. (5 points) Find the expected number of crimes will be solved in your town.

Answer 1

An FBI survey shows that about 80% of all property crimes go unsolved. In other words, the probability that any given crime is solved is 1-0.80=0.20

Let X be the number out of 3 crimes committed in the town that will be solved. Since the crimes are each deemed independent of each other, we can say that X has a Binomial distribution with parameters, number of trials (number of crimes committed in the the town) n=3, success probability (the probability that any given crime is solved) p=0.20.

The binomial probability that X=x crimes are solved is given by

Now we can get the values of P(x) for x=0,1,2,3 as below

Get this table

X	0	1	2	3
P(x)	0.512	0.384	0.096	0.008

Finally, get cumulative distribution F(x), by doing a running total of P(x)

a) ans:

X	0	1	2	3
P(x)	0.512	0.384	0.096	0.008
F(x)	0.512	0.896	0.992	1

b) The expected value of X is

ans: the expected number of crimes will be solved in your town is 0.60