# IE 227 RECITATION - 9 29.11.2013 1. A service station has both self ```IE 227 RECITATION - 9
29.11.2013
1. A service station has both self-service and full-service islands. On each island, there is a
single regular unleaded pump with two hoses. Let X denote the number of hoses being used on
the self-service island at a particular time, and let Y denote the number of hoses on the fullservice island in use at that time. The joint PMF of X and Y appear in the accompanying
tabulation.
y
P(x,y)
x
0
1
2
0
0.10
0.04
0.02
1
0.08
0.20
0.06
2
0.06
0.14
0.30
a) What is 𝑃(𝑋 = 1 𝑎𝑛𝑑 𝑌 = 1)=?
b) Compute 𝑃(𝑋 ≤ 1 𝑎𝑛𝑑 𝑌 ≤ 1).
c) Give a word description of event {𝑋 ≠ 0 𝑎𝑛𝑑 𝑌 ≠ 0}and compute the probability of this
event.
d) Compute the marginal PMF of X and of Y. Using 𝑃𝑥 (𝑥) what is P(X ≤ 1)?
e) Are X and Y independent random variables? Explain.
Answer: a) 𝑃(𝑋 = 1, 𝑌 = 1) = 𝑃𝑋,𝑌 (1, 1) = 0.20
b) 𝑃(𝑋 ≤ 1 𝑎𝑛𝑑 𝑌 ≤ 1) = 𝑃𝑋,𝑌 (0,0) + 𝑃𝑋,𝑌 (0, 1) + 𝑃𝑋,𝑌 (1, 0) + 𝑃𝑋,𝑌 (1, 1) = 0.42
c) At least one hose is in use at both islands. 𝑃(𝑋 ≠ 0 𝑎𝑛𝑑 𝑌 ≠ 0) = ⋯
2. Let X denotes the number of Canon digital cameras sold during a particular week by a certain
store. The PMF of X is
x
0
1
2
3
4
P(x) 0.1 0.2 0.3 0.25 0.15
Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let
Y denote the number of purchasers during this week who buy an extended warranty.
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a) What is 𝑃(𝑋 = 4, 𝑌 = 2)?
b) Calculate 𝑃(𝑋 = 𝑌).
c) Determine the joint PMF of X and Y and then the marginal PMF of Y.
Answer: a) P(𝑋 = 4, 𝑌 = 2) = 𝑃(𝑌 = 2|𝑋 = 4). 𝑃(𝑋 = 4)
You have four purchases and look for the buying 2 extended warranty in 4 purchases.
So Y|X ~ Bin(n=4, p=0.6)
P(𝑋 = 4, 𝑌 = 2) = 𝑃(𝑌 = 2|𝑋 = 4). 𝑃(𝑋 = 4) = (42)(0.6)2 (0.4)2 &times; (0.15) = 0.0518
b)…
3. Arrivals at the emergency rooms of two hospitals occur at average hourly rates of “α” and “&micro;”,
respectively. The joint PMF of the number of arrivals during any particular hour is
𝑃𝑋,𝑌 (𝑥, 𝑦) = 𝑒 −(𝛼+𝜇) (
𝛼𝑥𝜇𝑦
𝑥!𝑦!
) 𝑓𝑜𝑟 𝑥, 𝑦 = 0,1,2 … and zero otherwise.
a) Determine and identify each conditional distribution of Y given X=x.
b) Determine and identify each conditional distribution of X given Y=y.
c) What is the relationship between each conditional distribution of Y given X=x and the
marginal distribution of X.
Answer: a) 𝑃(𝑋 = 𝑥) = ∑~𝑦=0 𝑃𝑋,𝑌 (𝑋 = 𝑥, 𝑌 = 𝑦) = ∑~𝑦 𝑒 −(𝛼+𝜇) (
𝑒 −𝛼 𝛼 𝑥
𝑥!
𝛼𝑥𝜇𝑦
𝑥!𝑦!
)=
𝑒 −𝛼 𝛼 𝑥
𝑥!
∑~𝑦=0 𝑒 −𝜇
𝜇𝑦
𝑦!
=
𝑓𝑜𝑟 𝑥 = 0,1,2 …
𝑃(𝑌 = 𝑦|𝑋 = 𝑥) =
𝑃(𝑋 = 𝑥, 𝑌 = 𝑦) 𝑒 −𝜇 𝜇 𝑦
=
𝑓𝑜𝑟 𝑦 = 0,1,2 …
𝑃(𝑋 = 𝑥)
𝑦!
So Y|X~POI(&micro;)
b) 𝑃(𝑌 = 𝑦) = ∑~𝑦=0 𝑃𝑋,𝑌 (𝑋 = 𝑥, 𝑌 = 𝑦) =…
4. Suppose that 15 percent of the families in a certain community have no children, 20 percent
have one, 35 percent have two, and 30 percent have three; and suppose also that in each family,
each child is equally likely (independently) to be a boy or girl. If a family is chosen at random
from this community, then X, the number of boys, Y the number of girls, in this family.
a) Determine the joint PMF of X and Y.
b) Determine the marginal PMF of X and Y.
c) Find P(X &lt; Y).
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5. Among ten students, three are freshmen, four are sophomores and three are seniors.
Suppose that three of these students are selected at random with replacement. Let X, Y and Z
be the number of freshmen, sophomores and seniors, respectively, in the sample.
a) Identify and obtain the Joint PMF of X, Y and Z.
b) Use the result of part (a) and the FPF to determine and interpret each of P(X&gt;Y&gt;Z),
P(X≤Y&lt;Z) and P(X≤Z, Y≥Z).
c) Without doing any computations, identify and obtain the univariate marginal PMF’s of X,
Y and Z.
d) Without doing any computations, identify and obtain the PMF of each of the three
random variables X+Y, X+Z, and Y+Z.
e) Find P(1≤X+Z≤2).
Answer: a) X, Y, Z ~ Multinomial Distr. (n=3, px=0.3, py= 0.4, pz=0.3)
With joint PMF 𝑃𝑋,𝑌,𝑍 (𝑥, 𝑦, 𝑧)
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= (𝑥,𝑦,𝑧
) (0.3)𝑥 (0.4)𝑦 (0.3) 𝑧 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 + 𝑦 + 𝑧 =
3 𝑎𝑛𝑑 𝑥, 𝑦, 𝑧 ≥ 0
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b) 𝑃(𝑋 &gt; 𝑌 &gt; 𝑍) = 𝑃𝑋,𝑌,𝑍 (2,1,0) = (2,1,0
) (0.3)2 (0.4)1 (0.3)0 = 0.108
c)…
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