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Let X be the number of packages being mailed by a randomly selected customer at a...

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Question

Statistics

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

x1234

p(x)

0.10.40.20.3

(a)

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

X

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

X.

x11.522.533.54
P(x)

(b)

Refer to part (a) and calculate

P(X ≤ 2.5).

(c)

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

R0123
P(R)

(d)

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

P(X ≤ 1.5)?

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

x ≤ 1.5.]

Answer & Explanation Solved by verified expert
3.9 Ratings (712 Votes)

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
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