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\begin_layout Section
Practice Problems
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frametitle{Problems}
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\begin_layout Enumerate
The expected value of the height of a black bear is 5 feet.
 The expected value of the width of a black bear is 6 feet.
 You collected lots of data on the height and widths of bears during a long,
 boring field study.
 You are excited to know if the bears you study are like other bears that
 people have already studied.
 Before leaving the camp, you sent the data home to your assistant.
 Your research assistant mistakenly added the two columns of number together.
 You are in the 
\begin_inset Quotes eld
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publish or perish
\begin_inset Quotes erd
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 world, and you have to make something out of that data.
 Your numbers are the sum of height and width, you decide to call your new
 random variable the 
\begin_inset Quotes eld
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index of bear displacement (IBD).
\begin_inset Quotes erd
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\begin_deeper
\begin_layout Enumerate
What is the expected value of the IBD, supposing your bears are like other
 bears people have studied before? How do you figure that out when I don't
 give you a column of numbers?
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\begin_layout Enumerate
If I gave you a column of numbers for your bears, would it help you to calculate
 the expected value?
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On a conceptual level, how would you describe the difference between the
 expected value of the IBD and the sample average of your accidentally-added-tog
ether column of data?
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\end_deeper
\begin_layout Enumerate
Note one can convert from Celsius to Fahrenheit with 
\begin_inset Formula $F=32+\frac{9}{5}C$
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\begin_deeper
\begin_layout Enumerate
Your professor asks you to calculate the expected value of the temperature
 today.
 You get out some complicated probability model and work out the answer.
 The Expected Value of the temperature 18 Celsius.
 But the professor wanted the temperature in Fahrenheit, and you are too
 busy to go back and re-do all of the pdf calculations to get that for her.
 There's a shortcut to convert your result of 18C to Fahrenheit.
 How do you use it?
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\begin_layout Enumerate
This might force you to use your head.
 If the variance in Celsius is 4, what would be the variance of the same
 data presented in Fahrenheit? You have to use your head, but I'll give
 you a hint.
 The variance of a constant is always 
\begin_inset Formula $0$
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.
 So 
\begin_inset Formula $Var[32]=0$
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.
 Using that fact, I bet you can use the result stated in 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:VarWght"

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.
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I collected 1300 observations from a random process.
 I'm not sure which distribution was used, I forgot to write it down.
 But my teacher told me that E[x]=3 and Var[x]=10.
 The teacher told me to calculate the sample average and find out how certain
 she should be about that estimate.
 She said calculate Variance of the average.
 But I can't understand how to calculate the variance when I don't know
 to calculate 
\begin_inset Formula $Var[\bar{x}]$
\end_inset

.
 So I am completely stumped.
 Can you help?
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\begin_layout Enumerate
The sum of squared observations from a Normal distribution takes on the
 form of a Chi-Square distribution.
 Lets use R to check that out.
 Here's some code that use R to create 100 normal observations, and then
 squares them, and then calculates the cumulative sum.
 Then it makes a plot.
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<<problem2, fig=T, include=F, echo=T>>=
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xseq <- 1:100
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x <- rnorm(100)
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xsquare <- x*x
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chivalues <- cumsum(xsquare)
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plot(xseq, chivalues, type="l")
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@
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\begin_layout Enumerate
Think for a while about what that plotted line means.
 Then write a paragraph.
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\begin_layout Enumerate
Run that program over 10 or 20 times, compare the plots you get.
 What do the differences among them signify?
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\end_deeper
\begin_layout Enumerate
Lets beat up a Gamma variable for a while.
 Suppose I give you a Gamma distributed variable with shape = 
\begin_inset Formula $\alpha$
\end_inset

 and scale = 
\begin_inset Formula $\beta$
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.
 
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\begin_deeper
\begin_layout Enumerate
What is the Expected Value and the Variance?
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\begin_layout Enumerate
If 
\begin_inset Formula $\alpha$
\end_inset

 is replaced by a new shape parameter 
\begin_inset Formula $2/\beta$
\end_inset

, calculate 
\begin_inset Formula $E$
\end_inset

 and 
\begin_inset Formula $Var$
\end_inset


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\begin_layout Enumerate
Use R to sketch the pdf of Gamma(x; shape=2, scale=3)
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\begin_layout Enumerate
Use R to tell me the probability density of a particular value of x, say
 0.5.
 
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\begin_layout Enumerate
Use R to tell me the chance of observing a value between 0.5 and 1.4
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