Question

Let X be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of X is as follows.

x	1	2	3	4
p(x)	0.1	0.4	0.2	0.3

(a)

Consider a random sample of size n = 2 (two customers), and let

X

be the sample mean number of packages shipped. Obtain the probability distribution of

X.

x	1	1.5	2	2.5	3	3.5	4
P(x)

(b)

Refer to part (a) and calculate

P(X ≤ 2.5).

(c)

Again consider a random sample of size n = 2, but now focus on the statistic R = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of R. [Hint: Calculate the value of R for each outcome and use the probabilities from part (a).]

R	0	1	2	3
P(R)

(d)

If a random sample of size n = 4 is selected, what is

P(X ≤ 1.5)?

[Hint: You should not have to list all possible outcomes, only those for which

x ≤ 1.5.]

Answer 1

a) All the possible outcomes of n = 2 are:

Customer 1	Customer 2	Mean	R	Probability
1	1	1	0	0.01
1	2	1.5	1	0.04
1	3	2	2	0.02
1	4	2.5	3	0.03
2	1	1.5	1	0.04
2	2	2	0	0.16
2	3	2.5	1	0.08
2	4	3	2	0.12
3	1	2	2	0.02
3	2	2.5	1	0.08
3	3	3	0	0.04
3	4	3.5	1	0.06
4	1	2.5	3	0.03
4	2	3	2	0.12
4	3	3.5	1	0.06
4	4	4	0	0.09

The probability distribution is:

X	1	1.5	2	2.5	3	3.5	4
P(X)	0.01	0.08	0.2	0.22	0.28	0.12	0.09

b) P(X <= 2.5) = 0.01 + 0.08 + 0.20 + 0.22 = 0.51

c) The probability distribution from part a) is:

R	0	1	2	3
P (R)	0.3	0.36	0.28	0.06

d) Let's just list the outcomes with X <= 1.5:

C1	C2	C3	C4	Mean	Probability
1	1	1	1	1	0.0001
2	1	1	1	1.25	0.0004
1	2	1	1	1.25	0.0004
1	1	2	1	1.25	0.0004
1	1	1	2	1.25	0.0004
3	1	1	1	1.5	0.0002
1	3	1	1	1.5	0.0002
1	1	3	1	1.5	0.0002
1	1	1	3	1.5	0.0002
2	2	1	1	1.5	0.0016
2	1	2	1	1.5	0.0016
2	1	1	2	1.5	0.0016
1	2	2	1	1.5	0.0016
1	2	1	2	1.5	0.0016
1	1	2	2	1.5	0.0016

The probability is: 0.0121