Download Worksheet 14 Expected Value, Standard Deviation II

BUS/ST 350
LECTURE WORKSHEET #14
Expected Value, Standard Deviation II
Reiland
Name
ANSWERS
Lab
1. Kids
A couple plans to have children until they get a girl, but they agree that they will not
have more than 3 children even if they are all boys. (Assume boys and girls are
equally likely).
a. Let \ = the number of children that the couple will have. Find the probability
distribution for \ .
b. Find the expected number of children and the standard deviation of the number of
children that the couple will have.
c. Find the expected number of boys they will have.
ANSWERS: possible values of \ are "ß #ß $Þ
a. T Ð\ œ "Ñ œ :Ð"Ñ œ T Ð13<6Ñ œ "# à T Ð\ œ #Ñ œ :Ð#Ñ œ T Ð,9Cß
13<6Ñ œ "# ‡ "# œ "% à T Ð\ œ $Ñ œ :Ð$Ñ œ T Ð,9Cß ,9Cß ,9CÑ  T Ð,9Cß ,9Cß
13<6Ñ œ ")  ") œ "% Þ
\
" # $
:ÐBÑ "# "% "%
b. IÐ\Ñ œ "‡ "#  #‡ "%  $‡ "% œ " $%
WHÐ\Ñ œ Ð"  " $% Ñ# ‡ "#  Ð#  " $% Ñ# ‡ "%  Ð$  " $% Ñ# ‡ "%
œ Þ')(& œ Þ)#*# Þ
c. Let ] œ number of boys; values of ] are 0, 1, 2, 3.
T Ð] œ !Ñ œ :Ð!Ñ œ T Ð13<6Ñ œ "# à T Ð] œ "Ñ œ :Ð"Ñ œ T Ð,9Cß 13<6Ñ œ "% à
T Ð] œ #Ñ œ :Ð#Ñ œ T Ð,9Cß ,9Cß 13<6Ñ œ ") à T Ð] œ $Ñ œ :Ð$Ñ œ T Ð,9Cß ,9Cß
,9CÑ œ ") Þ
]
! " # $
:ÐCÑ "# "% ") ")
IÐ] Ñ œ !‡ "#  "‡ "%  #‡ ")  $‡ ") œ
(
)
Þ
2. Given independent random variables with means and standard deviations as shown in
the table, find the mean and standard deviation of each of these variables:
a. #]  #!
IÐ!Þ#&\  ] Ñ œ IÐ!Þ#&\Ñ  IÐ] Ñ
IÐ#]  #!Ñ œ IÐ#] Ñ  #!
!Þ#&IÐ\Ñ  IÐ] Ñ œ $# à
œ #IÐ] Ñ  #! œ %% à
WHÐ!Þ#&\  ] Ñ œ Z +<Ð!Þ#&\  ] Ñ
WHÐ#]  #!Ñ œ WHÐ#] Ñ
Z +<Ð!Þ#&\Ñ  Z +<Ð] Ñ
œ #WHÐ] Ñ œ '
œ Ð!Þ#&Ñ# Z +<Ð\Ñ  Z +<Ð] Ñ
b. $\
œ Ð!Þ#&Ñ# Ð"#Ñ#  Ð$Ñ#
IÐ$\Ñ œ $IÐ\Ñ œ #%! à
œ *  * œ %Þ#%$ Þ
WHÐ$\Ñ œ $WHÐ\Ñ œ $' Þ
d. \  &]
c. !Þ#&\  ]
IÐ\  &] Ñ œ IÐ\Ñ  &IÐ] Ñ
BUS/ST 350
Worksheet 13
œ )!  '! œ #! Þ
WHÐ\  &] Ñ œ Z +<Ð\  &] Ñ
œ Z +<Ð\Ñ  Z +<Ð&] Ñ œ
œ Ð"#Ñ#  #&Z +<Ð] Ñ
œ "%%  #&Ð*Ñ œ $'* œ "*Þ#" Þ
X
Y
page 2
Mean
80
12
SD
12
3
3. An insurance company estimates that on average it should make an annual profit of
$150 on each homeowner's policy written, with a standard deviation of $6000 (the
standard deviation is so large since most losses will be zero or very small, with a few
large losses; for example, not that many homes burn down but when they do, the loss
to the insurance company is large).
a. If the insurance company writes only two of these policies, what are the mean and
standard deviation of the annual profit? (assume policy profits are independent).
Let \" be the profit from policy 1; let \# be the profit from policy #Þ
IÐ\"  \# Ñ œ IÐ\" Ñ  IÐ\# Ñ œ $"&!  $"&! œ $$!! Þ
WHÐ\"  \# Ñ œ Z +<Ð\"  \# Ñ œ Z +<Ð\" Ñ  Z +<Ð\# Ñ
œ Ð'!!!Ñ#  Ð'!!!Ñ# œ (#!!!!!! œ $)%)&Þ#)
b. If the company writes 10,000 of these policies, what are the mean and standard
deviation of the annual profit?
Let \3 be the profit from policy 3, 3 œ "ß #ß ÞÞÞ ß "!ß !!!
IÐ\"  \#  â  \"!ß!!! Ñ œ  IÐ\3 Ñ œ "!ß !!!‡Ð$"&!Ñ œ $"ß &!!ß !!! Þ
"!ß!!!
WHÐ\"  \#  â  \"!ß!!! Ñ œ Z +<Ð\"  \#  â  \"!ß!!! Ñ
œ Z +<Ð\" Ñ  Z +<Ð\# Ñ  â  Z +<Ð\"!ß!!! Ñ
œ Ð'!!!Ñ#  Ð'!!!Ñ#  â  Ð'!!!Ñ# œ $'!ß !!!ß !!!ß !!!
œ $'!!ß !!! Þ
3œ"
c. Do you think the company is likely to be profitable if they sell 10,000 policies? (How
many standard deviations below the mean is $0?)
Since the expected value is $"ß &!!ß !!! and the standard deviation is $'!!ß !!!, the
D  =-9</ of $0 profit is D œ !"ß&!!ß!!!
œ  #Þ&, so $0 profit is 2.5 standard deviations
'!!ß!!!
below the mean. So if they sell 10,000 policies, the company is likely to be profitable.