Linear Combinations Exercises - Revision history

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Rb: Created page with " = Exercises =

Consider the joint probability distributions

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Created page with " = Exercises = <ol> <li>Consider the joint probability distributions <table border="1"> <tr class="odd"> <td align="left"></td> <td align="left"></td> <td align="lef..."

New page

= Exercises =

<ol>
<li>Consider the joint probability distributions
<table border="1">

<tr class="odd">
<td align="left"></td>
<td align="left"></td>
<td align="left"><math>Y:</math></td>
<td align="left"></td>
<td align="left"></td>
</tr>
<tr class="even">
<td align="left">Probabilities</td>
<td align="left"></td>
<td align="left"><math>-1</math></td>
<td align="left"><math>0</math></td>
<td align="left"><math>1</math></td>
</tr>
<tr class="odd">
<td align="left"><math>X:</math></td>
<td align="left"><math>-1</math></td>
<td align="left"><math>0.0</math></td>
<td align="left"><math>0.1</math></td>
<td align="left"><math>0.0</math></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><math>0</math></td>
<td align="left"><math>0.2</math></td>
<td align="left"><math>0.3</math></td>
<td align="left"><math>0.3</math></td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><math>1</math></td>
<td align="left"><math>0.0</math></td>
<td align="left"><math>0.0</math></td>
<td align="left"><math>0.1</math></td>
</tr>

</table>

<table border="1">

<tr class="odd">
<td align="left"></td>
<td align="left"></td>
<td align="left"><math>Y:</math></td>
<td align="left"></td>
<td align="left"></td>
</tr>
<tr class="even">
<td align="left">Probabilities</td>
<td align="left"></td>
<td align="left"><math>-1</math></td>
<td align="left"><math>0</math></td>
<td align="left"><math>1</math></td>
</tr>
<tr class="odd">
<td align="left"><math>X:</math></td>
<td align="left"><math>-1</math></td>
<td align="left"><math>0.02</math></td>
<td align="left"><math>0.10</math></td>
<td align="left"><math>0.08</math></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><math>0</math></td>
<td align="left"><math>0.06</math></td>
<td align="left"><math>0.30</math></td>
<td align="left"><math>0.24</math></td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><math>1</math></td>
<td align="left"><math>0.02</math></td>
<td align="left"><math>0.10</math></td>
<td align="left"><math>0.08</math></td>
</tr>

</table>

In each case,
<ol>
<li>find the marginal probability distributions of <math>X</math> and <math>Y</math>;</li>
<li>find out whether <math>X</math> and <math>Y</math> are independent.</li></ol>
</li>
<li>You are an investment consultant employed by an investor who intends to invest in the stock market or in a deposit account with a building society. The percentage annual rate of return for the stock market is denoted by the random variable <math>S</math>. For simplicity we assume that this rate of return will be one of four values: -10%, 0%,10% or 20%. The annual rate of interest on the deposit account (denoted <math>R</math>) will be 4%, 6% or 8%. From previous experience, you believe that the joint probability distribution for these variables is:
<table border="1">

<tr class="odd">
<td align="left"></td>
<td align="left"></td>
<td align="center"><math>S:</math></td>
<td align="center"></td>
<td align="center"></td>
<td align="center"></td>
</tr>
<tr class="even">
<td align="left">Probabilities</td>
<td align="left"></td>
<td align="center"><math>-10</math></td>
<td align="center"><math>0</math></td>
<td align="center"><math>10</math></td>
<td align="center"><math>20</math></td>
</tr>
<tr class="odd">
<td align="left"><math>R:</math></td>
<td align="left"><math>4</math></td>
<td align="center"><math>0</math></td>
<td align="center"><math>0</math></td>
<td align="center"><math>0.1</math></td>
<td align="center"><math>0.1</math></td>
</tr>
<tr class="even">
<td align="left"></td>
<td align="left"><math>6</math></td>
<td align="center"><math>0</math></td>
<td align="center"><math>0.1</math></td>
<td align="center"><math>0.3</math></td>
<td align="center"><math>0.1</math></td>
</tr>
<tr class="odd">
<td align="left"></td>
<td align="left"><math>8</math></td>
<td align="center"><math>0.1</math></td>
<td align="center"><math>0.1</math></td>
<td align="center"><math>0.1</math></td>
<td align="center"><math>0</math></td>
</tr>

</table>

<ol>
<li><ol>
<li>Find the marginal probability distributions for <math>R</math> and <math>S</math>. What is the overall probability that the rate of return for the stock market will be positive? What is the probability that the rate of return for the stock market will exceed that from the building society deposit account?</li>
<li>Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment?</li></ol>
</li>
<li>Calculate the (population) covariance and (population) correlation between <math>R</math> and <math>S</math>. How would you interpret the value of the correlation?</li>
<li><ol>
<li>One proposal you make to the investor is to split her savings equally between the stock market and the building society account. Find (using any appropriate method) the mean and variance of the random variable
<math>A=0.5R+0.5S.</math></li>
<li>Why might the investor prefer the proposed new 50/50 strategy to the simple strategies of investing all her money in the building society or in the stock market?</li></ol>
</li></ol>
</li>
<li>The random variables <math>X</math> and <math>Y</math> have <math>\mu _{X}=10</math>, <math>\sigma_{X}=3,\mu _{Y}=-1,\sigma _{Y}=4</math>.
<ol>
<li>Find the mean and standard deviation of <math>V=X+Y</math> when
<ol>
<li><math>X</math> and <math>Y</math> are independent;</li>
<li><math>\sigma _{XY}=-8</math>.</li></ol>
</li>
<li>Find the mean and standard deviation of
<math>W=3X-2Y+8</math>
when <math>X</math> and <math>Y</math> are independent.</li></ol>
</li>
<li>In the manufacture of disposable syringes, the manufacturing process produces cylinders of diameter <math>X_{1}\sim N\left( 20.2,0.04\right) </math> and plungers of diameter <math>X_{2}\sim N\left( 19.7,0.0225\right)</math>.
<ol>
<li>If the components are combined so that <math>X_{1}</math> is independent of <math>X_{2}</math>, what proportion of plungers will not fit?</li>
<li>Suppose now that the components are not independently combined, but that larger plungers tend to be combined with larger cylinders, leading to <math>Cov\left[ X_{1},X_{2}\right] =0.02</math>. What proportion of plungers will not fit now?</li></ol>
</li></ol>

= Footnotes =

← Older revision		Revision as of 10:24, 11 September 2014
Line 2:		Line 2:

	= Exercises =		= Exercises =
		+
		+	Final solutions to these questions can be seen in the {curly brackets}. Fully worked solutions are available from the following clips: [http://youtu.be/3fDEGa7o6So?hd=1 Q1], [http://youtu.be/7r2yZGBtKYY?hd=1 Q2], [http://youtu.be/eeuVNUqo3Ww?hd=1 Q3] and [http://youtu.be/t2LFixEF1-s?hd=1 Q4].
		+

	<ol>		<ol>

@@ Line 150: / Line 150: @@
 </li></ol>
 </li>
-<li><p>The random variables <math>X</math> and <math>Y</math> have <math>\mu _{X}=10</math>, <math>\sigma_{X}=3,\mu _{Y}=-1,\sigma _{Y}=4</math>.</p>
+<li><p><math>[L1,L2]</math> The random variables <math>X</math> and <math>Y</math> have <math>\mu _{X}=10</math>, <math>\sigma_{X}=3,\mu _{Y}=-1,\sigma _{Y}=4</math>.</p>
 <ol>
 <li><p>Find the mean and standard deviation of <math>V=X+Y</math> when</p>
 <ol>
-<li><p><math>X</math> and <math>Y</math> are independent;</p></li>
+<li><p><math>X</math> and <math>Y</math> are independent; {9;5}</p></li>
-<li><p><math>\sigma _{XY}=-8</math>.</p></li></ol>
+<li><p><math>\sigma _{XY}=-8</math>. {9;3}</p></li></ol>
 </li>
 <li><p>Find the mean and standard deviation of</p>
 <p><math>W=3X-2Y+8</math></p>
-<p>when <math>X</math> and <math>Y</math> are independent.</p></li></ol>
+<p>when <math>X</math> and <math>Y</math> are independent. {40;12.0416}</p></li></ol>
 </li>
-<li><p>In the manufacture of disposable syringes, the manufacturing process produces cylinders of diameter <math>X_{1}\sim N\left( 20.2,0.04\right) </math> and plungers of diameter <math>X_{2}\sim N\left( 19.7,0.0225\right)</math>.</p>
+<li><p><math>[L2]</math>In the manufacture of disposable syringes, the manufacturing process produces cylinders of diameter <math>X_{1}\sim N\left( 20.2,0.04\right) </math> and plungers of diameter <math>X_{2}\sim N\left( 19.7,0.0225\right)</math>.</p>
 <ol>
-<li><p>If the components are combined so that <math>X_{1}</math> is independent of <math>X_{2}</math>, what proportion of plungers will not fit?</p></li>
+<li><p>If the components are combined so that <math>X_{1}</math> is independent of <math>X_{2}</math>, what proportion of plungers will not fit? {0.0228}</p></li>
-<li><p>Suppose now that the components are not independently combined, but that larger plungers tend to be combined with larger cylinders, leading to <math>Cov\left[ X_{1},X_{2}\right] =0.02</math>. What proportion of plungers will not fit now?</p></li></ol>
+<li><p>Suppose now that the components are not independently combined, but that larger plungers tend to be combined with larger cylinders, leading to <math>Cov\left[ X_{1},X_{2}\right] =0.02</math>. What proportion of plungers will not fit now? {app. 0}</p></li></ol>
 </li></ol>
 = Footnotes =

@@ Line 140: / Line 140: @@
 <li><ol>
 <li><p>Find the marginal probability distributions for <math>R</math> and <math>S</math>. What is the overall probability that the rate of return for the stock market will be positive? {0.7} What is the probability that the rate of return for the stock market will exceed that from the building society deposit account? {0.7}</p></li>
-<li><p>Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment? R:6.2,1.96; S:10,80</p></li></ol>
+<li><p>Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment? {R:6.2,1.96; S:10,80}</p></li></ol>
 </li>
-<li><p>Calculate the (population) covariance and (population) correlation between <math>R</math> and <math>S</math>. How would you interpret the value of the correlation? -7.6; -0.60693</p></li>
+<li><p>Calculate the (population) covariance and (population) correlation between <math>R</math> and <math>S</math>. How would you interpret the value of the correlation? {-7.6; -0.60693}</p></li>
 <li><ol>
 <li><p>One proposal you make to the investor is to split her savings equally between the stock market and the building society account. Find (using any appropriate method) the mean and variance of the random variable</p>
 <p><math>A=0.5R+0.5S.</math></p>
-<p>8.1;16.69</p></li>
+<p>{8.1;16.69}</p></li>
 <li><p>Why might the investor prefer the proposed new 50/50 strategy to the simple strategies of investing all her money in the building society or in the stock market?</p></li></ol>
 </li></ol>

@@ Line 89: / Line 89: @@
 <ol>
 <li><p>find the marginal probability distributions of <math>X</math> and <math>Y</math>;</p></li>
-<li><p>find out whether <math>X</math> and <math>Y</math> are independent Solution: Distribution 1: no; Distribution 2: yes.</p></li></ol>
+<li><p>find out whether <math>X</math> and <math>Y</math> are independent {Solution: Distribution 1: no; Distribution 2: yes}.</p></li></ol>
 </li>
 <li><p><math>[L1,L2]</math> You are an investment consultant employed by an investor who intends to invest in the stock market or in a deposit account with a building society. The percentage annual rate of return for the stock market is denoted by the random variable <math>S</math>. For simplicity we assume that this rate of return will be one of four values: -10%, 0%,10% or 20%. The annual rate of interest on the deposit account (denoted <math>R</math>) will be 4%, 6% or 8%. From previous experience, you believe that the joint probability distribution for these variables is:</p>
@@ Line 139: / Line 139: @@
 <ol>
 <li><ol>
-<li><p>Find the marginal probability distributions for <math>R</math> and <math>S</math>. What is the overall probability that the rate of return for the stock market will be positive? 0.7 What is the probability that the rate of return for the stock market will exceed that from the building society deposit account? 0.7</p></li>
+<li><p>Find the marginal probability distributions for <math>R</math> and <math>S</math>. What is the overall probability that the rate of return for the stock market will be positive? {0.7} What is the probability that the rate of return for the stock market will exceed that from the building society deposit account? {0.7}</p></li>
 <li><p>Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment? R:6.2,1.96; S:10,80</p></li></ol>
 </li>

@@ Line 4: / Line 4: @@
 <ol>
-<li><p>Consider the following two joint probability distributions. Distribution 1:</p>
+<li><p><math>[L1,L2]</math> Consider the following two joint probability distributions. Distribution 1:</p>
 <table border="1">
@@ Line 89: / Line 89: @@
 <ol>
 <li><p>find the marginal probability distributions of <math>X</math> and <math>Y</math>;</p></li>
-<li><p>find out whether <math>X</math> and <math>Y</math> are independent.</p></li></ol>
+<li><p>find out whether <math>X</math> and <math>Y</math> are independent Solution: Distribution 1: no; Distribution 2: yes.</p></li></ol>
 </li>
-<li><p>You are an investment consultant employed by an investor who intends to invest in the stock market or in a deposit account with a building society. The percentage annual rate of return for the stock market is denoted by the random variable <math>S</math>. For simplicity we assume that this rate of return will be one of four values: -10%, 0%,10% or 20%. The annual rate of interest on the deposit account (denoted <math>R</math>) will be 4%, 6% or 8%. From previous experience, you believe that the joint probability distribution for these variables is:</p>
+<li><p><math>[L1,L2]</math> You are an investment consultant employed by an investor who intends to invest in the stock market or in a deposit account with a building society. The percentage annual rate of return for the stock market is denoted by the random variable <math>S</math>. For simplicity we assume that this rate of return will be one of four values: -10%, 0%,10% or 20%. The annual rate of interest on the deposit account (denoted <math>R</math>) will be 4%, 6% or 8%. From previous experience, you believe that the joint probability distribution for these variables is:</p>
 <table border="1">
@@ Line 139: / Line 139: @@
 <ol>
 <li><ol>
-<li><p>Find the marginal probability distributions for <math>R</math> and <math>S</math>. What is the overall probability that the rate of return for the stock market will be positive? What is the probability that the rate of return for the stock market will exceed that from the building society deposit account?</p></li>
+<li><p>Find the marginal probability distributions for <math>R</math> and <math>S</math>. What is the overall probability that the rate of return for the stock market will be positive? 0.7 What is the probability that the rate of return for the stock market will exceed that from the building society deposit account? 0.7</p></li>
-<li><p>Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment?</p></li></ol>
+<li><p>Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment? R:6.2,1.96; S:10,80</p></li></ol>
 </li>
-<li><p>Calculate the (population) covariance and (population) correlation between <math>R</math> and <math>S</math>. How would you interpret the value of the correlation?</p></li>
+<li><p>Calculate the (population) covariance and (population) correlation between <math>R</math> and <math>S</math>. How would you interpret the value of the correlation? -7.6; -0.60693</p></li>
 <li><ol>
 <li><p>One proposal you make to the investor is to split her savings equally between the stock market and the building society account. Find (using any appropriate method) the mean and variance of the random variable</p>
-<p><math>A=0.5R+0.5S.</math></p></li>
+<p><math>A=0.5R+0.5S.</math></p>
 <li><p>Why might the investor prefer the proposed new 50/50 strategy to the simple strategies of investing all her money in the building society or in the stock market?</p></li></ol>
 </li></ol>