Math 2b, Winter 2009: Solutions to Homework 4
1. A new elevator in a large hotel is designed to carry about 30 people, with a total weight of up
to 5000 lbs. More than 5000 lbs. overloads the elevator. The average weight of guests at this
hotel is 150 lbs., with an SD of 55 lbs. Suppose 30 of the hotel’s guests get into the elevator.
Assuming the weights of these guests are independent random variables, what is the chance of
overloading the elevator? Give your approximate answer as a decimal.
Solution.
Let
X
1
, . . . , X
30
be random variables representing the weights of the 30 guests who get
into the elevator, and let
S
=
∑
30
i
=1
X
i
. The
X
i
’s are identically distributed independent random
variables with mean 150 and standard deviation 55, which means
S
is a random variable with
mean 150
·
30 = 4500 and standard deviation 55
√
30. There are enough people in the elevator to
use the Central Limit Theorem to say that
S
is approximately normally distributed. Thus the
probability of overloading the elevator is approximately the probability of being at least
5000

4500
55
√
30
≈
1
.
660
standard deviations above the mean in a normal distribution. This is given by
1

Φ
(1
.
660)
≈
0
.
048,
so there is roughly a 5% chance of overloading the elevator.
2. Suppose
X
and
Y
are independent random variables such that
EX
4
= 2,
EX
2
= 1,
EY
2
= 1,
and
EY
= 0. Let
W
=
X
2
Y
. Calculate
EW
,
EW
2
,
SD
(
W
).
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 Winter '08
 Makarov,N
 Math, Differential Equations, Statistics, Equations, Normal Distribution, Probability, Standard Deviation, Variance, Probability theory, hX +Y

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