Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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.