Question

The library records the number of books checked out by each person over the course of one day, with the following result:

Number of books checked out	Probabilities
0	20%
1	45%
2	20%
3	10%
4	5%

State the random variable

Draw a histogram of the number of booked checked out by each patron over the course of one day

Find the mean, variance, and standard deviation

What is the probability that a patron will check out at least one book?

What is the probability will take out no more than three books?

Is it unusual for a patron to check out more than three books? Why?

Answer 1

Let X is a random variable shows the number of booked checked out by each patron over the course of one day. Following table shows the probability distribution of X:

Number of books checked out, X	Probabilities, P(X=x)
0	0.2
1	0.45
2	0.2
3	0.1
4	0.05
Total	1

Following is the histogram:

4 3 2 1 0 55453525150 2 2 0 0 0 2

Following table shows the calculations:

Number of books checked out, X	Probabilities, P(X=x)	xP(X=x)	x^2*P(X=x)
0	0.2	0	0
1	0.45	0.45	0.45
2	0.2	0.4	0.8
3	0.1	0.3	0.9
4	0.05	0.2	0.8
Total	1	1.35	2.95

So,

$E(X)=\sum xP(X=x)=1.35$
$Var(X)=\sum x^{2}P(X=x)-\left [ \sum xP(X=x) \right ]^{2}=2.95-1.35^{2}=1.1275$

$\sigma=\sqrt{\sum x^{2}P(X=x)-\left [ \sum xP(X=x) \right ]^{2}}=1.0618$

------------------

The probability that a patron will check out at least one book is

$P(X\geq 1)=1-P(X=0)=1-0.20=0.80$

The probability will take out no more than three books is

$P(X\leq 3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=0.20+0.45+0.20+0.10= 0.95$

The probability will take out more than three books is

P(X >3) = P(X=4) = 0.05

Since above probability is not less than 0.05 so it is not unusual for a patron to check out more than three books.